basic of geodesy
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Geodesy :- Geo - Earth desy - The study of
“The study of the Earth”
Geodesy is the science of the measurement and mapping of the
earth’s surface
Merriam-Webster:a branch of applied mathematics concerned with the determination of the size and shape of the earth and the exact positions of points on its surface and with the description of variations of its gravity field.
• Geometrical geodesy is concerned with describing locations in terms of geometry. Consequently, coordinate systems are one of the primary products of geometrical geodesy.
•Physical geodesy is concerned with determining the Earth’s gravity field, which is necessary for establishing heights.
•Satellite geodesy is concerned with using orbiting satellites to obtain data for geodetic purposes.
Types of Geodesy
R
ψ
ΔG
R = ΔG /Ψ
Spherical Model of the Earth
Eratosthenes had observed that on the day of the summer solstice (20-22 June), the midday sun shone to the bottom of a well in the Ancient Egyptian city of Swenet (known in Greek as Syene).
Eratosthenes Egypt about 240 BC
Spherical Model of the Earth
Syene
Alexandria
500mi
•He knew that at the same time, the sun was not directly overhead at Alexandria; instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12'). •He also knew that Alexandria and Syene were 500 miles apart•To these observations, Eratosthenes concluded that the circumference of the earth was 50 x 500 miles, or 25000 miles.
Spherical Model of the Earth
•The accepted value along the equator is 24,902 miles, but, if you measure the earth through the poles the value is 24,860 miles•He was within 1% of today’s accepted value•Eratosthenes' conclusions were highly regarded at the time, and his estimate of the Earth’s size was accepted for hundreds of years afterwards.
Spherical Model of the Earth
How Do We Define the Shape of the Earth?
We think of the earth as a sphere
It is actually a spheroid, slightly
larger in radius at the equator than at the
poles
The Ellipsoid
An ellipse is a mathematical figure which is defined by
Semi-Major Axis (a) andSemi-Minor Axis (b) orFlattening (f) = (a - b)/a
It is a simple geometrical surface
Cannot be sensed by instruments
b
a
Ellipsoid or Spheroid
O
X
Z
Ya ab
Rotational axis
Rotate an ellipse around an axis
Selection of the Spheroid is what determines the size of the Earth
The Real Earth (The Geoid)
EuropeN. America
S. America Africa
Topography
An ellipsoidal-earth model is no longer tenable at a high level of accuracy. The deviation of the physical measurements refer from the ellipsoidal model can no longer be ignored.
The geoid anomaly is the difference between the geoid surface and a reference ellipsoid
The Real Earth (Geoid)
What is the Geoid?
• “The equipotential surface of the Earth’s gravity field which best fits, in the least squares sense, global mean sea level.”
• Can’t see the surface or measure it directly.
• Modeled from gravity data.
Ellipsoid and Geoid
Ellipsoid• Simple Mathematical
Definition• Described by Two Parameters• Cannot be 'Sensed' by
Instruments Geoid• Complicated Physical
Definition• Described by Infinite Number
of Parameters• Can be 'Sensed' by
Instruments
Earth surface
EllipsoidSea surface
Geoid
Since the Geoid varies due to local anomalies, we must approximate it with a ellipsoid
Ellipsoid and Geoid
O1
EuropeN. America
S. America Africa
TopographyN
Ellipsoid and Geoid
Which ellipsoid to choose ?
O2
O1
EuropeN. America
S. America Africa
NTopography
N
Ellipsoid and Geoid
Common Ellipsoid
The best mean fit to the Earth
EuropeN. America
S. America Africa
NTopography
A = 6,378,137.000 m1/f = 298.2572236
Unfortunately, the density of the earth’s crust is not uniformly the same. Heavy rock, such as an iron ore deposit, will have a stronger attraction than lighter materials. Therefore, the geoid (or any equipotential surface) will not be a simple mathematical surface.
Ellipsoid and Geoid Heights
Heighting
•The equipotential surface is forced to deform upward while remaining normal to gravity. This gives a positive geoid undulation.
•Conversely, a mass shortage beneath the ellipsoid will deflect the geoid below the ellipsoid, causing a negative geoid undulation.
EllipsoidEllipsoid
PP
HH
GeoidGeoid
hhTopographyTopography
h = H + Nh = H + N EllipsoidEllipsoid
hhPP TopographyTopography
HH
GeoidGeoidNNN = Geoidal Separation
H = Height above Geoid(~Orthometric Height)
h = Ellipsoidal height
Heighting
Orthometric Height (h) “ perpendicular vertical distance between the geoid and land surface”
Heighting The height difference
between ellipsoid and geoid is called the geoidal undulation
To obtain orthometric heights, the geoidal undulation must be accounted for
EllipsoidEllipsoid
PP
HH
GeoidGeoidNN
N = Geoidal Separation
hhTopographyTopography
Latitude and Longitude
Lines of latitude are called “parallels”
Lines of longitude are called “meridians”
The Prime Meridian passes through Greenwich, England
Latitude and Longitude in N. America
90 W120 W 60 W
30 N
0 N
60 N
Length on Meridians and Parallels
0 N
30 N
Re
Re
RR
A
BC
(Lat, Long) = (, )
Length on a Meridian:AB = Re (same for all latitudes)
Length on a Parallel:CD = R Re Cos(varies with latitude)
D
Any set of numbers, usually in sets of two or three, used to determine location relative to other locations in two or three dimensions
Coordinate systems
Global Cartesian Coordinates (x,y,z)
O
X
Z
Y
GreenwichMeridian
Equator
•
•A system for the whole earth
•Non manageable and difficult to relate to other locations when translated to two dimensions
•The z-coordinate is defined as geometrically
Geographic Coordinates (, z)
• Latitude () and Longitude () defined using an ellipsoid, an ellipse rotated about an axis
• Elevation (z) defined using geoid, a surface of constant gravitational potential
Origin of Geographic Coordinates
(0,0)Equator
Prime Meridian
Coordinate System
(o,o)(xo,yo)
X
Y
Origin
A planar coordinate system is defined by a pairof orthogonal (x,y) axes drawn through an origin
() (x, y)Map Projection
Coordinate System
