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Lecture 7 – More Gravity and GPS Processing
GISC-3325
5 February 2009
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Update
• Next class and lab will be on OPUS and GPS processing using OPUS.
• First exam on 12 February 2009.– Chapters 1-4 and all lectures and labs.– Will take place during Lab period at CI229
• Lab 3 – Automated GPS processing and review due on 10 February 2009.
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Reminder
• You are responsible for the material listed as REQUIRED ADDITIONAL READING in the Class 5 section of the web page.
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Gravity field and steady-state Ocean Circulation Explorer
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Earth’s Gravity Field from Space• Satellite data was
used for global models– Only useful at
wavelengths of 700 km or longer
• Shorter wavelength data from terrestrial or marine gravity of varying vintage, quality and geographic coverage
Terrestrial and marine gravity data in NGS data base.
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Note the discontinuity at the shoreline.
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http://www.ngs.noaa.gov/GRAD-D
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Gravity
• Static gravity field – Based on long-term average within Earth
system
• Temporally changing component– Motion of water and air– Time scale ranges from hours to decades.
• Mean and time variable gravity field affect the motion of all Earth space vehicles.
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• Is the vector sum of gravitational and centrifugal acceleration.
• The actual acceleration of gravity varies from place to place, depending on latitude, altitude, and local geology.
• By agreement among physicists, the standard acceleration of gravity gn is defined to be exactly 9.80665 meters per second per second (m s-2), or about 32.174 05 feet per second per second.
Gravitational Attraction
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• Magnitude of the potential is the work that must be done by gravity to move a unit mass from infinity to the point of interest.
• Is dependent on position within the gravitation field.
Gravitational Potential
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• Surface having constant gravity potential NOT constant gravity.– Also known as level surfaces or
geopotential surfaces.
• Surfaces are perpendicular at all points of the plumb line (gravity vector).
• A still lake surface is an equipotential surface. – It is not horizontal but curved.
Equipotential Surfaces
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Geoid
• The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level.
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Image from Featherstone, W, “Height Systems and Vertical Datums” Spatial Science, Vol 51 No. 1, June 2006
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• They are closed continuous surfaces that never cross one another.
• They are formed by long radius arcs. Generally without abrupt steps.
• They are convex everywhere.
Properties of equipotential surfaces
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Geopotential Number
• C (geopotential number) – is a value derived from the difference in gravity potential between an equipotential surface of interest and the geoid.
• Represents the work required to move a 1kg mass from the geoid to the geopotential surface at the point of interest.
• C is numerically similar to the elevation of the point in meters.
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Leveling Issues
• Height – The distance measured along a perpendicular
between a point and a reference surface.
• Raw leveled heights– Non-unique. Depending on the path taken a
different height will be determined for the same point. They have NO physical relevance.
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Equipotential Surfaces
HCHA
Reference Surface (Geoid)
HAC hAB + hBC
Observed difference in orthometric height, H, depends on the leveling route.
AC
B Topography
hAB
h = local leveled differences
Leveled Height vs. Orthometric Height
= hBC
H = relative orthometric heights
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• Vertical distance from the geoid to the point of interest. (along curved plumb line).
• Orthometric height (H) may be determined from H = geopotential number / mean gravity along plumb line– We use assumptions about mass density to
estimate mean gravity.
Orthometric Height
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Geopotential Numbers
• Geopotential numbers– Unique, path-independent– Geopotential number (C) at a point of interest
(p) is the difference between the potential on the geoid and potential at the point.
• C = W0-Wp
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Review of Height Systems• Helmert Orthometric• NAVD 88
• local gravity field ( )• single datum point• follows MSL
g
HC
g
C
g H
g gh
G H
0 0 4 2 41
23
.
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How are they computed?
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Dynamic Heights
• Takes the geopotential number at a point and divides it by a constant gravity value.
• The constant value used in the US is the normal gravity value at 45 degrees N latitude – 980.6199 gals
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CC area Dynamic Heights
NAVD 88 and dynamic heights differ by only 1 mm at this station.
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Dynamic Heights
NAVD 88 = 2026.545 m
DYN Hght= 2023.109
Difference = 3.436 m
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Error condition: input position is inconsistent with predicted one.
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Dynamic Height Problems
• Dynamic height corrections applied to spirit-leveled height differences can be very large (several meters) if the chosen gravity value is not representative for the region of operation.
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Orthometric correction
• Occurs because equipotential surfaces are not parallel.
• In general, not parallel in a N-S direction but they are parallel in an E-W direction.
• It is computed as a function of mean orthometric height and latitude of section.
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Leveled v Ortho. Height Diffs
• Due to non-parallelism of level surfaces the differences between published NAVD 88 points do not correspond to the leveled height difference.
• Therefore the subtraction of the published NAVD 88 heights on the NGS datasheets will NOT agree with the leveled difference.
• This is problem mostly in high elevations.
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An example from Colorado
• The difference between published NAVD 88 heights is 402.156 m.
• The leveled difference is 402.085 m.