Gravimetric Network Adjustments

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Gravimetric Network Adjustments

• J.-P. Barriot
• BGI
• MicroGravimetry School, Lanzarote, 23-28 October,
  2005

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Why gravimetric network adjustments ?

• To determine time drifts of relative gravimeters
• (Scintrex 0.5 mgal/day, Lacoste 0.5 mgal/month)
• To determine calibration factors (F 10E-6 to
  10E-3)

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Network Equations Example of typical form
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1
Network Equation Two observations g1 and g2 at
different places at times t1 and t2
Network Equation Difference of these two
observations
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Network Equations Difference of the two
observations at times t1 and t2
Network Equations Difference of the two
observations at times t1 and t2 (rewriting)
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Network Equations Difference of the two
observations at times t1 and t2 (rewriting)
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Network Equations 3 observations with loop
2
1 and 1
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Drift b ? 0.5 mgal/day Calibration F ? 10E-3
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(No Transcript)
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But if we replace g1 and g2 by g1 and g2 in
such a way that
Then
We cannot determine g, but gcte, with cte
arbitrary gt rank deficiency in G
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But also
2
1 and 1
3
10
2
1 and 1
3
gt interplay between calibration F and time
drift b gt instability
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Usual Least-Squares gt
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But
does not exist !
Because we cannot determine g, but gcte, with
cte arbitrary gt rank deficiency in G
Besides gt interplay between calibration F and
time drift b gt instability problem
ill-conditioned
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Solution 1 Adding Equations, for example
2
1 and 1
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Sufficient in principle, but ill-conditioning
(interplay between F and b)
VERY GOOD !
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Solution 2 Truncated inverse (singular values
decomposition SVD)
Pb choice of ?i
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Solution 2 (continued) Truncated inverse
(singular values decomposition SVD)
Permitted range of the solution around
zero, including gravity g, drifts b and
calibration factor F

• Choice of truncation level i
• norm(solution) not too small, not too big
• AND
• - norm (residuals) not too small, not too big

GOOOOD !
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Solution 3 Tikonov approximate inverse
Regularisation parameter
Permitted range of the solution around
zero, including gravity g, drifts b and
calibration factor F
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Solution 3 (continued) Tikonov approximate
inverse

• Choice of parameter ?
• norm(solution) not too small, not too big
• AND
• - norm (residuals) not too small, not too big

GOOOOD !
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Solution 4 Iterative inverse
Iterative process
Example
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Solution 4 (continued) Iterative inverse
But
MATHEMATICAL TRICK REALLY BAAAAD!
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A small example, just to illustrate
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Network Geometry station numbering
Leg 1
Leg 2
2
1
3
4
5
6
7
8
10
9
11
12
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Network Geometry true gravity in mgal
Leg 1
Leg 2
10
7
10
7
7
14
14
7
10
7
10
7
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Network Geometry measurement times along leg 1
in min
Leg 1
30
60
90
0/360
120
150
180
210
270
240
300
330
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Network Geometry measurement times along leg 2
in min
Leg 2
150
0/360
180
330
300
210
120
30
90
60
240
270
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Network true gravity in mgal along stations
mgal
station number
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Singular values sv0 no added Eq. sv1 one added
Eq. (g(1)7) sv2 two added Eq. (g(1)7 and
g(6)14)
Log10
Singular value index
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Solution with added Eqs. red crosses true
gravity values blue no added Eq. cyan one added
Eq. (g(1)7) green two added Eq. (g(1)7 and
g(6)14)
mgal
station number
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Solution with truncated SVD red true gravity
values blue trucation 4 green truncation
8 cyan truncation 12
mgal
station number
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Solution with Tikonov red true gravity
values green alpha1E-20 cyan alpha1E-3 blue
alpha1E3
mgal
station number
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Solution with Tikonov choice of regularisation
parameter red norm of the solution green norm
of residuals
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Example of iterated solution red true gravity
values green iterated solution
mgal
station number
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Determination of calibration factor F and time
drifts b1 and b2
TRUE VALUES F 0.00112000000000 b1
0.0003472222222 b2 0.0002777777777
one added eq. F -1.00000000061203
b10.00000000000023 b2-0.00000000000004 two
added eq. F0.00101936318604
b10.00035321519057 b20.00028507612674
truncated 8 F-0.00000004414763
b10.00000000000000 b20.00000000000000 Tikonov
F-0.00000221038955 b10.00019957846307
b20.00012944781684
iterated F-0.56655269405289
b10.00015294430885 b20.00012343968305
Time drifts in mgal/min
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SUMMARY
Solution 1 Adding Equations VERY GOOD
Solution 2 Truncated Inverse GOOD
Solution 3 Tikonov Inverse GOOD
Solution 4 Iterative Inverse VERY BAD
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THE END
Reverend Lejay, founder of BGI