Summary of Last Presentation
• Reference system
• Conventional reference system
• Reference frame
– Space fixed
– Earth fixed: CTS
• ITRF
• WGS84
• NAD
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 1
Problem with Ellipsoid Heights
• They are new: easily obtained since the
80s with GPS, not accurate enough
before. By comparison, triangulation exists
since around 1500.
• They do not represent what most users
want which is to see that “water flows
downhill!” => need potential based height.
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 2
Basic Definitions
• Gravitational potential of the Earth=Newtonian potential
generated by the mass distribution within the Earth:
V(P)  G 
B
 (Q)

z
dB Q
l
Q
G  Newton's constant 6.67x10-11 m3kg1sec2
P
• Gravity potential of the Earth:

1
W P   V(P)  ω 2 x 2  y 2
2
  angular velocityof theEarth 0.729212x1
0-4 sec1
• Gravity vector:
Catherine LeCocq
SLAC

O
i
y
j
x
 r r 
g(P)  W(P) G    P 3 Q   (Q) dB  ω2 xi  yi 
 

USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 3
Gravity Vector-Astronomical Quantities
plumb line
z
level surfaces
W = const.
P
g
y
Λ
Φ
x
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 4
Geoid and orthometric Heights
•
The geoid is a particular level surface chosen to be close to the average
surface of the oceans: WP  W0
P2
HP2 (>0)
HP1 (<0)
oceanic floor
P1
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 5
Approximations of V
• Zero level:
μ
V
r
μ  GM  3.986x1014 m3sec2
M  totalmass of theEarthincluding atmosphere
• First order:
2
μ
1 
R 3 2
V  1  J 2    sin   
r 
2 
 r  2
  sphericallatitude
R  mean radius of theEarth
J 2  103
• Leveled ellipsoid
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 6
Ellipsoidal Coordinates for Potential U

1
UP   Ve(P)  ω 2 x 2  y 2
2

u
 (P)  U(P)
β
F1
P
F2
U depends on 4 constants:
- ω is known
- U0 is computed by imposing: mass of the ellipsoid = mass of Earth
- a and E are derived from satellite observations choosing Earth J2
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 7
Different Types of Geoid
– Current NGS definition of geoid: The equipotential surface of the
Earth's gravity field which best fits, in a least squares sense, global
mean sea level
– The definition of the geoid is complicated by the permanent deformation
of the Earth caused by the presence of the Sun and the Moon.
Consideration of these permanent tidal effects has led to the definition
of three types of geoids and three types of reference ellipsoids [Ekman,
1989, 1995; Rapp et al., 1991; Bursa, 1995a]. The three geoids are
described as follows:
1. Tide-free (or nontidal)—This geoid would exist for a tide-free Earth
with all (direct and indirect) effects of the Sun and Moon removed.
2. Mean—This geoid would exist in the presence of the Sun and the
Moon (or, equivalently, if no permanent tidal effects are removed).
3. Zero—This geoid would exist if the permanent direct effects of the
Sun and Moon are removed, but the indirect effect component
related to the elastic deformation of the Earth is retained.
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 8
Tidal Effects
• Gravitational Potential
• Tidal effects:
W
GM
l
– permanent / periodic
– direct / indirect
z
P
B
l
r
O
d
y
• Nomenclature
– no effect => free
x
– only direct-permanent => zero-tide
– all permanent =>mean
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 9
Geoid in the US
• NGS Definition: The equipotential surface of the Earth's gravity
field which best fits, in a least squares sense, global mean sea level.
• NGS Warnings: Even though we adopt a definition, that does not
mean we are perfect in the realization of that definition. For
example, altimetry is often used to define "mean sea level" in the
oceans, but altimetry is not global (missing the near polar regions).
As such, the fit between "global" mean sea level and the geoid is not
entirely confirmable. Also, there may be non-periodic changes in sea
level (like a persistent rise in sea level, for example). If so, then
"mean sea level" changes in time, and therefore the geoid should
also change in time. These are just a few examples of the difficulty
in defining "the geoid".
• Latest models: GEOID03, USGG2003, GEOID99, G99SSS
• Presentations: http://www.ngs.noaa.gov/GEOID/PRESENTATIONS/
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 10
Leveling
terrain
geoid
• The process of
precise leveling is
to measure height
differences and to
sum these to get
the heights of other
points.
ellipsoid
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 11
Different Height Systems
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 12
Orthometric Height
• How to compute gravity inside the Earth?
– Assume a simple terrain model: Bouguer
plate (standard density 2.67g/cm3)
– Perform the following steps:
• Remove the Bouguer plate
• Do the free air anomaly
• Restore the Bouguer plate
– Final formula:
• Helmert heights:
Catherine LeCocq
SLAC
g p  gQ  0.0848( H P  HQ )
C
H
g  0.0424 H
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 13
Pizzetti Projection
P

H
earth’s
surface
h
Po
N
Q
Catherine LeCocq
SLAC
Qo
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
geiod
ellipsoid
Height Systems 14
Deflections of the Vertical
ellipsoidal
surface
normal
astronomic
normal
CTP
α
Ze
φ
Za
CTP
ACT
P
ФCTP
P2
α
T
P1
equipotential
surface
through P1
T
ε
Ze
η
ξ
θ
terrestrial equator
Za
φ
CTP
ACTP
ξ  Φ CTP  
η  Λ CTP  λ  cos
  ξcosα  ηsinα
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 15
Methods for Geoid Computations
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 16
Three Components of the Geoid
NGM = long wavelength calculated from a geopotential model
Ng = medium wavelength computed with Stokes
NT = terrain correction
Catherine LeCocq
SLAC
USPAS, Cornell University
Large Scale Metrology of Accelerators
June 27 - July 1, 2005
Height Systems 17