Improved formulae for consistent combination of geometric...
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Improved formulae for consistent combination of geometric and orthometric
heights and their rates
C. Kotsakis and I. Tsalis
Department of Geodesy and Surveying Aristotle University of Thessaloniki
Thessaloniki, Greece
Presented at the IAG International Symposium on Gravity, Geoid and Height Systems (GGHS 2012)
9-12 October 2012, Venice, Italy
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Objective
Revise the theoretical constraint for geometrical and
physical heights beyond the simplified model
What will we gain ?
Standardized and geodetically meaningful de-trending for the joint analysis of heterogeneous heights
Frame-consistent modeling of heterogeneous vertical velocities
Direct combination of quasi-geoids with orthometric heights !
0h N H 0h N H
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Γεωειδές H
Fundamental relationship
( ) ( )o o
W W
LVDo o oW W W
h N Hg g
Ref. Ellipsoid
h LVD
oW W
oW W
Topo
H
N
Expresses in linearized form the vertical separation between equipotential surfaces
g
(Wo) refers to the (almost constant) gravity on the geoid
h and N should refer to the same GRS (ellipsoid + frame)
DoV is ignored but the model error is negligible (< 1 mm)
Note:
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Revised relationship
Instead of the (non-determinable) true orthometric heights,
we shall employ the actual Helmert orthometric heights.
helm
helmgH H
g
( )
oW
helmhelm helm oWg g
h N H Hg g
Height-correlated residuals even with error-free data !
Using the formula:
we obtain the following condition:
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Revised relationship (cont’d)
Looks like a similarity transformation for different VRFs
( )
oW
helmhelm helm oWg g
h N H Hg g
2 1 1 VRF VRF VRFH H H
Vertical scale factor (λ)
Zero-height vertical offset (μ)
Reflect the ‘vertical datum disturbance’ between the underlying height frames
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0 500 1000 1500 2000 2500-2
-1.5
-1
-0.5
0
0.5
1
Helmert ortho height (m)
e (m
)
0 500 1000 1500 2000 2500-2
-1.5
-1
-0.5
0
0.5
1
Helmert ortho height (m)
EGM08
Example (1542 Greek GPS/lev BMs)
/ GPS N helm helmH H H e
Linear theoretical trend
ˆ 43.1 cm
3ˆ 0.12 10
ˆ 72.6 cm
3ˆ 0.57 10
EGM96
HG
PS
/N –
Hhelm
(m
)
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0 500 1000 1500 2000 2500-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Helmert ortho height (m)
No
rma
lize
d e
0 500 1000 1500 2000 2500-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
Helmert ortho height (m)
Example (1542 Greek GPS/lev BMs)
Nonlinear theoretical trend
/
GPS N helm
helm helm
H He
H H
EGM08 EGM96
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Conclusion #1
Rigorous constraint between GPS/geoid and
Helmert orthometric heights
( )
/
oW
GPS N helm helm oWH H H
g
The parameters λ and δWo reflect the systematic differences
between the underlying VRFs
Geodetically meaningful comparison of heterogeneous height data
A constant conventional value g
(Wo) is required for the LS inversion of
the above model
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Use of quasi-geoid models
g
N H
helm
helm helm oWgh H H
Based on the well-known formula:
and after few justifiable approximations (< 1 mm), we obtain the following condition:
which is ‘similar’ to the formula
( )
oW
helmhelm helm oWg g
h N H Hg g
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Conclusion #2
( )1/
1/
oWhelm
helmohelm
h N H gH W
h H
Is there any value in using the second relationship for the joint analysis of quasi-geoid models with
Helmert ortho heights ?
Rigorous constraints for heterogeneous heights
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Example (1542 Greek GPS/lev BMs)
h - N
EGM08 - H
helm (12.45 ± 0.75) × 10-5 -8.55 ± 0.05 m2 s-2
h - ζ
EGM08 - H
helm (2.13 ± 0.73) × 10-5 -8.35 ± 0.05 m2 s-2
ˆoW
max min mean σ
Residuals from model A
0.481 -0.473 0.000 0.130
Residuals from model B
0.465 -0.464 0.000 0.127
all values in m ( ) 2 9.78 m soW
g
Model A
Model B
Estimated VRF transformation parameters
Statistics of adjusted residuals
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Example (20 Swiss EUVN-DA BMs)
h - N
EGM08 - H
EVRF07 (-2.37 ± 3.66) × 10-5 -6.18 ± 0.41 m2 s-2
h - ζ
EGM08 - H
EVRF07 (-15.53 ± 3.04) × 10-5 -5.87 ± 0.34 m2 s-2
ˆoW
max min mean σ
Residuals from model A
0.160 -0.154 0.000 0.090
Residuals from model B
0.110 -0.156 0.000 0.075
all values in m ( ) 2 9.78 m soW
g
Model A
Model B
Estimated VRF transformation parameters
Statistics of adjusted residuals
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Joint modeling of heterogeneous vertical velocities
By taking the time derivative of the (static) constraint
helm
helm helm oWgh H H
helm
helm oWHh H g
(and after some lengthy derivations) we οbtain the
following condition
negligible terms
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Joint modeling of heterogeneous vertical velocities
By taking the time derivative of the (static) constraint
helm
helm oWHh H g
(and after some lengthy derivations) we οbtain
the following condition
negligible terms
Effect of total gravity
variation on the
Earth’s surface
0.1 μGal/yr
1.0 μGal/yr
10 μGal/yr
100 m 10-5 10-4 10-3
1000 m 10-4 10-3 10-2
ghelmH
Numerical effect in mm/yr
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Joint modeling of heterogeneous vertical velocities
By taking the time derivative of the (static) constraint
helm
helm oWHh H g
(and after some lengthy derivations) we οbtain
the following condition
negligible terms
Required datum-related term !
Its role is similar to that of the shift-rate parameters in TRF
velocity transformation
x x x v v t
It is zero provided that there is no-net-vertical-motion between the underlying VRFs !
2 1VRF VRFH H
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t
Ref. Ellipsoid
H
LVDoW W
oW
H = 0
N
Dynamic
geoid
Ref. Ellipsoid
t'
Vertical crustal motion + gravity field variation
?
Example
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t
Ref. Ellipsoid
H
LVDoW W
oW
H = 0
N
Dynamic
geoid
Ref. Ellipsoid
H' ≠ H
LVDoW W
H = 0
t'
Example
Vertical crustal motion + gravity field variation
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t
Ref. Ellipsoid
H
LVDoW W
oW
H = 0
N
Dynamic
geoid
(GRACE)
Ref. Ellipsoid
H' ≠ H
LVDoW W
oW
H = 0
N' ≠ N
Dynamic
geoid
(GRACE)
t'
Example
Vertical crustal motion + gravity field variation
oW UoW U
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The meaning of oW
It is associated with the temporal evolution of the zero-
height equipotential reference surfaces
It is affected by the realization of:
the dynamic vertical datum → H
helm(t)
the time-dependent geoid model → N(t) or ζ(t)
It is a fundamental ‘datum parameter’ that needs to be
a priori constrained when computing a dynamic VRF from
the optimal combination of multiple data sources
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A useful theoretical constraint
Based on the previous condition, we have
helm
ohelm helm
Wg h H
h h H H h
which can be used for estimating the gravity-to-height ratio
from heterogeneous vertical velocities
Repeated gravity measurements are not required !
Useful for validation of Earth models and comparison with geophysical predictions for various physical processes (e.g. PGR)
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Conclusion – Future work
A revision of the simplified model
is necessary in the context of modern VRF theory and practice
A general conventional re-formulation has been presented in this study
Numerical tests with heterogeneous vertical velocities need to be performed (under our new formulation) over key areas, i.e. Canada, Fennoscandia
0h N H 0h N H
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