geodesy tutorial

GEODESY

WHAT IS GEODESY

GEO – EARTH DESY – STUDY GEODESY – THE STUDY OF EARTH

IT IS THE SCIENCE OF THE MEASUREMENT AND MAPPING OF EARTH

OBLATE SPHEROID

W E

P’

P

a

b

On the International Spheroid, 1924 :

a = 3444.054 n miles.

b = 3432.458 n miles.

; Since Earth is flattened at the Poles. Infact the 1924 Spheroid is a close mathematical match with WGS 72 and WGS 84 (for Navstar GPS).

FLATTENING OF THE EARTH

• Since the Earth is flattened at the poles, it can be quantified as f = a – b

a• Therefore the International Spheroid 1924, it

will = (3444.054 – 3432.458) / 3444.054 = 11.596 / 3444.054 = 1 / 297

ECCENTRICITY• In mathematics, the eccentricity, denoted e or , is

a parameter associated with every conic section.

• It can be thought of as a measure of how much the conic section deviates from being circular.

• The eccentricity of an ellipse which is not a circle is greater than zero but less than 1.

ECCENTRICITY

• The eccentricity of an ellipse is strictly less than 1. • For any ellipse, let a be the length of its

semi-major axis and b be the length of its semi-minor axis.

• We define a number of related additional concepts (only for ellipses):

ECCENTRICITY

• When a point moves so that it’s distance from a fixed point S (the focus) is always in a constant ratio e (less than unity) to it’s perpendicular distance from a fixed straight line AB (the directrix), the locus of M is called an ellipse of eccentricity e

O

Y

X

M

SD

A

B

C

a

b

ECCENTRICITY

• MS = eMC, e may be then defined as

e = (2f – f2)½ O

Y

X

M

SD

A

B

C

a

b

GEODETIC AND GEOCENTRIC LATITUDE

• MK is the tangent to point M on the spheroid.

• If the normal to this tangent cuts the OA (equatorial plane) in L, then MLA is the Geodetic Latitude of M.

• The angle MOA is called the Geocentric Latitude.

• These two are related– Tan θ = b2/a2 (tan ø) = (1 – f2) tan ø = (1 – e2) tan ø

øθ

M

O L A

K

• GEODETIC VERSUS GEOCENTRIC LATITUDE• IT IS IMPORTANT TO NOTE THAT GEODETIC

LATITUDE IS DIFFERENT FROM GEOCENTRIC LATITUDE.GEODETIC LATITUDE IS DETERMINED BY THE ANGLE BETWEEN THE NORMAL OF THE SPHEROID AND THE PLANE OF THE EQUATOR, WHEREAS GEOCENTRIC LATITUDE IS DETERMINED AROUND THE CENTRE (SEE FIGURE). UNLESS OTHERWISE SPECIFIED LATITUDE IS GEODETIC LATITUDE.

The reduced or parametric latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the vertical projection of a point P on the ellipsoid at latitude The reduced or parametric latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the vertical projection of a point P on the ellipsoid at latitude The reduced or parametric latitude, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the vertical projection of a point P on the ellipsoid at latitude

THE SAME POSITION ON A SPHEROID HAS A DIFFERENT ANGLE FOR LATITUDE DEPENDING ON WHETHER THE ANGLE IS MEASURED FROM THE NORMAL (ANGLE Α) OR AROUND THE CENTER (ANGLE Β). NOTE THAT THE "FLATNESS" OF THE SPHEROID (ORANGE) IN THE IMAGE IS GREATER THAN THAT OF THE EARTH; AS A RESULT, THE CORRESPONDING DIFFERENCE BETWEEN THE "GEODETIC" AND "GEOCENTRIC" LATITUDES IS ALSO EXAGGERATED.

PARAMETRIC LATITUDE

• The figure shows a meridional section of a spheroid WPE, it’s polar axis is OP and it’s shape and size are defined by ‘a’ and ‘b’.

• WBE is a sphere drawn with radius ‘a’.

• M is a point on the spheroid with Geodetic Lat ø.

• If we extend HM (parallel to OP) to cut the circle WBE at U. The radius OU makes angle β with the x axis. This is the parametric or reduced latitude of the point M. The parametric latitude is often used on the spheroid for calculation of long distances.

øβ

M

O L E

B

H

b

a

P U

W

LENGTH OF ONE MINUTE OF LATITUDE

Considering the spheroid, the length of one minute of latitude is calculated using an advanced formula that includes eccentricity as well. (Ref Page 44 / 45 of BR 45(3))

Geodesic. The Geodesic is analogous to the great circle on a spheroid. It is the shortest distance between two points on the spheroidal Earth.

3. Geoid surface where strength of gravity equals that at mean sea level

variations in rock density and topography causes deviations up to 100 m

irregular (geoid) vs regular (ellipsoid)

MAJOR MODEL OF GEODESY

MAJOR MODEL OF GEODESY

rises over continents, depressed in oceanic areas

ELLIPSOID

• The shape of an ellipsoid (of revolution) is determined by the shape parameters of that ellipse which generates the ellipsoid when it is rotated about its minor axis.

• The semi-major axis of the ellipse, a, is identified as the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b, is identified with the polar distances (from the centre).

• These two lengths completely specify the shape of the ellipsoid but in practice geodesy publications classify reference ellipsoids by giving the semi-major axis and the inverse flattening, 1/f,

• The flattening, f, is simply a measure of how much the symmetry axis is compressed relative to the equatorial radius.

ORTHOMORPHIC PROJECTION• Conformal projections preserve local shape.• Graticule lines on the globe are perpendicular. • To preserve individual angles describing the spatial

relationships, a conformal projection must show graticule lines intersecting at 90 degree angles on the map. This is accomplished by maintaining all angles.

• The drawback is that the area enclosed by a series of arcs may be greatly distorted in the process. No map projection can preserve shapes of larger regions