**Surveying Heights**

**By : Surveying & Geomatic Department**

Definition of heights

-Ellipsoidal height (geometric)

-Orthometric height (potential field based)

Shape of equipotential surface: Geoid for Earth

Methods for determining heights

Overview on advanced search on the Internet

Ellipsoidal heights

Calculation of ellipsoid heights from Cartesian XYZ.

The ellipsoid height is the distance along the normal to the reference ellipsoid from the surface of the ellipsoid to the point who height is being calculated.

While the geometric quantities, geodetic latitude and longitude are used for map mapping and terrestrial coordinates in general; ellipsoidal height is almost never used.

Ellipsoidal heights

Local and Global Ellipsoid

Orthometric heights are heights above an equipotential surface

The equipotential surface is called the geoid and corresponds approximately to mean sea level (MSL).

The correspondence is approximately because MSL is not an equipotential surface because of forces from dynamic ocean currents

Orthometric heights

Mean Sea Level (MSL)

Ocean tides also need to be considered but this can be averaged over time (signal is periodic with semi-diurnal, diurnal and long period tides. Longest period tide is 18.6 years)

Another major advantage of MSL is that is has been monitored at harbors for many centuries in support of ocean going vessels

Also poses a problem because dredging of harbors can change the tides.

Land-locked countries had to rely on other countries to tell them the heights at the border.

MSL is reasonably consistent around the world and so height datums differ by only a few meters (compared to hundreds of meters for geodetic latitude and longitude.

Height determination

The figure on the next page shows how the technique called leveling is used to determine heights.

In a country there is a primary leveling network, and other heights are determined relative to this network.

The primary needs to have a monument spacing of about 50 km.

Orthometric height of hill is

Dh1+Dh2+Dh3

N is Geoid Height. Line at bottom is ellipsoid

In ideal cases, elevation angles at both ends are measured at the same time. This helps cancel atmospheric refraction errors.

The distance D can be many tens of kilometers.

D is determined either by triangulation or by electronic distance measurement (EDM).

The heights of the instruments, called theodolites, above the ground point must be measured.

How Were Geodetic Datums Established?

Although the difference between ellipsoidal and orthometric height allows the geoid height to be determined, this method has only be been used since GPS became available.

Determining the geoid has been historical done using surface gravity measurements and satellite orbits.

Satellite orbit perturbations reveal the forces acting on the satellite which if gravity is the only effect is the first derivative of the potential.

Trigonometric Leveling

Geoid height

The long wavelength part of geoid (greater than 1000km) is now determined from satellite orbit perturbations.

The <1000km wavelength use surface gravity and solve a boundary value problem where the derivative of the function which satisfies Laplace’s equation is given on the boundary, and the value of the function is needed.

Most of the great mathematicians worked on field theory trying to solve the Earth boundary value problem (Laplace, Legendre, Green, Stokes)

The standard method of converting gravity measurements to geoid height estimates is called Stokes method.

This field is called physical geodesy

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