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12.540 Principles of the Global
Positioning System
Lecture 18
Prof. Thomas Herring
04/17/02 12.540 Lec 18 2
Mathematical models in GPS
? Review assignment dates (updated on class
web page)
– Paper draft due Mon April 29
– Homework 3 due Fri May 03
– Final class is Wed May 15. Oral presentations of
papers. Each presentation should be 15-20
minutes,with additional time for questions.
? Next three lectures:
– Mathematical models used in processing GPS
– Processing methods used
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Mathematical models used in GPS
? Models needed for millimeter level positioning
? Review of basic estimation frame:
– Data (phase and pseudorange) are collected at a sampling
interval (usually 30-sec) over an interval usually a multiple of
24-hours. Typically 6-8 satellites are observed simultaneously
– A theoretical model is constructed to model these data. This
model should be as complete as necessary and it uses apriori
values of the parameters of the model.
– An estimation is performed in which new values of some of
the parameters are determined that minimize some cost
function (e.g., RMS of phase residuals).
– Results in the form of normal equations or covariance
matrices may be combined to estimate parameters from many
days of data (Dong D., T. A. Herring, and R. W. King, Estimating
Regional Deformation from a Combination of Space and Terrestrial
Geodetic Data, J. Geodesy, 72, 200–214, 1998.)
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Magnitude of parameter adjustments
? The relative size of the data noise to effects of a
parameter uncertainty on the observable determines
in general whether a parameter should be estimated.
? In some cases, certain combinations of parameters
can not be estimated because the system is rank
deficient (discuss some examples later)
? How large are the uncertainties in the parameters that
effect GPS measurements?
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Magnitudes of parameter adjustments
? Major contributions to GPS measurements:
– Pseudorange data: Range from satellite to receiver, satellite
clock and receiver clock (±10 cm)
– Phase data: Range from satellite to receiver, satellite clock
oscillator phase, receiver clock oscillator phase and number of
cycles of phase between satellite and receiver (±2 mm)
? Range from satellite to receiver depends on
coordinates of satellite and ground receiver and
delays due to propagation medium (already
discussed).
? How rapidly do coordinates change? Satellites move
at 1 km/sec; receivers at 500 m/s in inertial space.
? To compute range coordinates must in same frame.
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Parameter adjustment magnitudes
? Already discussed satellite orbital motion:
Parameterized as initial conditions (IC) at specific time
and radiation model parameters.
? For pseudo range positioning, broadcast ephemeris is
often adequate. Post-processed orbits (IGS) ±3-5 cm
(may not be adequate for global phase processing).
? Satellites orbits are easiest integrated in inertial
space, but receiver coordinates are nearly constant in
an Earth-fixed frame.
? Transformation between the two systems is through
the Earth orientation parameters (EOP). Discussed in
Lecture 4.
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EOP variations
? If analysis is near real-time, variations in polar
motion and UT1 will need to be estimated.
? After a few weeks, these are available from
IERS (±0.2 mas of pole position, 0.05 ms
UT1) in the ITRF2000 no-net-rotation system.
? For large networks, normally these
parameters are re-estimated. Partials are
formed by differentiating the arguments of the
rotation matrices for the inertial to terrestrial
transformation.
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Position variations in ITRF frame
? The International Terrestrial Reference Frame (ITRF)
defines the positions and velocities of ~1000 locations
around the world (GPS, VLBI, SLR and DORIS).
? Frame is defined to have no net rotation when
motions averaged over all tectonic plates.
? However, a location on the surface of the Earth does
not stay at fixed location in this frame: main
deviations are:
– Tectonic motions (secular and non-secular)
– Tidal effects (solid Earth and ocean loading)
– Loading from atmosphere and hydrology
? First are normally accounted for.
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Solid Earth Tides
? Solid Earth Tides are the deformations of the
Earth caused by the attraction of the sun and
moon. Tidal geometry
M*
P l
R
r
ψ
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Solid Earth Tide
? The potential at point P U=GM*/l
? We can expand 1/l as:
? For n=0; U
0
is GM*/R and is constant for the whole
Earth
? For n=1; U
1
=GM*/R
2
[r cosψ]. Taking the gradient of
U
1
; force is independent of position in Earth. This
term drives the orbital motion of the Earth
1
l
=
1
R
2
?2Rrcosψ+ r
2
=
r
n
R
n+1
n=0
∞
∑
P
n
(cosψ)
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Solid Earth Tide
? The remaining terms are the tidal potential, U
T
.
? Second form is often referred to as the “vector” tide
model (convenient if planetary ephemeredes are
available)
U
T
=
GM *
R
r
R
?
?
?
?
?
?
n=2
∞
∑
n
P
n
(cosψ)
U
T
= GM *
1
l
?
1
R
?
R.r
R
3
?
?
?
?
?
?
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Solid Earth Tide
? The work of Love on tides showed that the
response of the (spherical) Earth is dependent
on the degree n of the tidal deformation and
that:
d
r
=
h
n
g
U
T
? e
r
Radial
d
t
=
l
n
g
?U? e
t
=
l
n
g
?U
T
?θ
?
θ +
l
n
gsinθ
?U
T
?λ
?
λ Tangential
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Solid Earth Tide
?h
n
and l
n
are called Love numbers (also a k
n
for the change in potential, needed for orbit
integration).
? For the moon r/R=1/60 and for the Sun
r/R=1/23,000: Most important tidal terms are
2nd degree harmonics: k
2
=0.3; h
2
=0.609;
l
2
=0.085
? Expand the second harmonic term in terms of
θ, λ of point and θ’, λ’ extraterrestrial body
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Solid Earth Tide:
? Resultant expansion gives characteristics of
tides:
U
T
=
GM * r
R
3
2
cos
2
θ?
1
2
?
?
?
?
?
?
?
?
?
3
2
cos
2
θ'?
1
2
?
?
?
?
?
?
→ Long period
+ cosθsinθcosθ'sinθ' cos(λ?λ')
[]
→Diurnal
+
1
4
sin
2
θsin
2
θ' cos2(λ?λ')
[]
→ Semidiurnal
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Solid Earth Tide
? Magnitude: h
2
(GM*/R)r=26.7 cm.
? Long period tides: 0 at ±35 degree latitude
? Diurnal tides: Max at mid-latitudes
? Semidiurnal tides: zero at poles
? The planetary positions θ’, λ’ have periodic variations
that set the primary tidal frequencies.
? Major lunar tide M2 has a variation with period of
13.66 days (1/2 lunar period)
? Additional consideration: Presence of fluid core affects
the tides. Largest effect is ?h
2
=-0.089 at 1
cycle/sidereal day
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Series expansion
? The tidal potential can be expanded in series in terms
of:
lm, ls - Longitude of moon and sun,
ω - Argument of lunar perigee,
GST - Greenwich sidereal time
? The other system used with tides is Doodson’s
arguments:
τ - Time angle in lunar days;
s, h - Mean longitude of Sun and Moon
p, p1 - Long of Moon's and Sun’s perigee
N' - Negative of long of Moon's Node
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Earth tides:
? The Fortan routines earth_tide.f and gst_jd.f
compute the tidal displacement at any location
on the Earth. (This routine uses numerical
derivatives for the tangential components.
Analytic derivatives are not that difficult to
derive.)
? (The const_param.h file contains quantities
such as pi).
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Ocean tides
? The ocean tides also load the solid Earth causing and
additional tidal like signal in the Earth.
? At the temporal frequencies of the tides, both systems
behave linearly and so the temporal frequencies of the
response is the same.
? For the solid Earth tides, the spatial frequency
response is also linear but no so for the ocean tides.
?The P
2
forcing of the ocean tides, generates many
spherical harmonics in the ocean response and thus
the solid earth response has a complex spatial
pattern.
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Ocean tides
? The solution to the ocean tidal loading
problem requires knowledge of the ocean tide
potential (the level of the tides) and the
loading response of the Earth.
? The loading problem has a similar solution to
the standard tidal problem but in this case
load Love numbers, denoted k
n
’, h
n
’ and l
n
’are
used.
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Load Love numbers
? The load love numbers depend on the elastic
properties of the Earth (deduced from seismic
velocities)
n-h
n
’nl
n
’-nk
n
’
1 0.290 0.113 0
2 1.001 0.059 0.615
3 1.052 0.223 0.585
4 1.053 0.247 0.527
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Load signal calculations
? For simple homogenous Earth, the Love numbers
depend on rigidity of the Earth
? Load signals can be computed by summing all the
spherical harmonics.
? An alternative is a Green’s function approach (Farrell,
1972) in which the response to a point load is
computed (the point load is expanded in spherical
harmonics)
? The Green’s function can then be convolved with a
surface load to compute the amount of deformation.
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Ocean loading magnitudes
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Loading signals
? In addition to ocean tidal loading, any system
that loads the surface will cause loading
deformations.
? Main sources are:
– Atmospheric pressure loading (~0.5 mm/mbar).
Often short period, but annual signals in some parts
of the world.
– Surface water loading (~0.5 mm/cm of water).
More difficult to obtain load data
? In some locations, sediment expansion when
water added (eg. LA basin)
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Basic loading effect
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
?H (mm/mbar)
?R (mm/mbar)
?
H (mm/mbar)
Radius (deg)
Displacements due 10
o
radius load (2200 km diameter)
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Example: Penticton Canada
-20
-10
0
10
20
1999.95 2000.00 2000.05 2000.10 2000.15
JPL
MIT
Atm Load
DRAO Height (mm)
Year
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Summary
? Tides and loading
– All high-precision GPS analyses account for solid-
Earth tides; most account for ocean tidal loading
– Loading effects for the atmosphere, surface water
and non-tidal ocean loading are not commonly
directly applied because inputs are uncertain.
– Atmospheric pressure loading could be routinely
applied soon (data sets are high quality)
– Gravity mission GRACE might recover surface
loads well enough to allow these to be applied
routinely (current research topic).