Transcript Geoid determination by least-squares collocation using GRAVSOFT

Geoid determination by
least-squares collocation using
GRAVSOFT
C.C.Tscherning, University of Copenhagen, 2005-01-28
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Purpose:
Guide to gravity field modeling, and especially to geoid
determination, using least-squares collocation (LSC).
DATA

GRAVITY FIELD MODEL

EVERYTHING
=
Height anomalies, gravity anomalies, gravity
disturbances, deflections of the vertical, gravity
gradients, spherical harmonic coeffients
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Quasi-geoid:
Important:
the term geoid = the quasi-geoid,
i.e. the height anomaly evaluated at the surface of
the Earth.
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Gravsoft
The use of the GRAVSOFT package of FORTRAN
programs will be explained.
A general description of the GRAVSOFT programs
are given in
http://cct.gfy.ku.dk/gravsoft.txt
which is updated regularly when changes to the
programs have been made.
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FORTRAN 77
• All programs in FORTRAN77.
• Have been run on many different computers under
many different operating systems.
• Available commercially, but free charge if used for
scientific purposes.
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General methodology
• General methodology for (regional or local)
gravity field modelling :
• Transform all data to a global geocentric geodetic
datum (ITRF99/GRS80/WGS84), (but NO tides,
NO atmosphere) GEOCOL
• “geoid-heights” must be converted to height
anomalies N2ZETA
• Use the remove-restore method.
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Remove-restore method
• The effect of a spherical harmonic expansion and
of the topography is removed from the data and
• subsequently added to the result. GEOCOL, TC,
• TCGRID
• This is used for all gravity modelling methods
including LSC.
• This will produce what we will call residual data.
(Files with suffix *.rd).
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Covariance
Determine at statistical model (a covariance
function) for the residual data in the region in
question.
EMPCOV, COVFIT
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Select
Make a homogeneous selection of the data to be
used for geoid determination using rule-ofthumbs for the required data density, SELECT
X
o
o
X
x o
xo
o
o
x
x
If many data select those with the smallest error X
Selection of points O closest to the middle. 6 points
selected
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Errors
• check for gross-errors (make histograms and
contour map of data), GEOCOL
• verify error estimates of data, GEOCOL.
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Gravity field approximation and datum
– Determine using LSC a gravity field approximation,
including contingent systematic parameters such as
height system bias N0. GEOCOL
– Compute estimates of the height-anomalies and their
errors. GEOCOL
– If the error is too large, and more data is available add
new data and repeat.
–
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Restoring and checking.
• Check model, by comparison with data not used to
obtain the model. GEOCOL.
• Restore contribution from Spherical Harmonic
model and topography. GEOCOL, TC.
• Convert height anomalies to geoid heights if
needed N2ZETA.
• The whole process can be carried through using
the GRAVSOFT programs
• Compare with results using other methods !
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Test Data
• GRAVSOFT includes data from New Mexico,
USA, which can be used to test the programs and
procedures. (Files: nmdtm, nmfa, nmdfv etc.)
• They have here been used to illustrate the use of
the programs.
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Anomalous potential.
• The anomalous gravity potential, T, is equal to the
difference between the gravity potential W and the
so-called normal potential U,
T = W-U.
• T is a harmonic function, and may as such be
expanded in solid spherical harmonics, Ynm

T( ,  , r)  GM
n

n 2 m  n
•
Cnm Ynm ( ,  , r)
GM is the product of the gravitational constant and the mass of the Earth and the fuly
normalized spherical harmonic coefficients.
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Coordinates used.
GEOCOL accepts geocentric, geodetic and Cartesian
(X,Y,Z) coordinates but output only in geodetic.
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Solid spherical harmonics.
Ynm (P)  Ynm ( ,  , r) 
n
a
 cos m ,0  m  n
n  1 Pnm (sin  ) 
r
 sin| m|  , n  m  0
• where a is the semi-major axis and Pnm the
Legendre functions.
• We have orthogonality:
/2 
1
Ynm ( ,  , R ) Yij ( ,  , R ) cos( )d d


4  / 2 
 1, n  i, m  j

 0, n  i  m  j
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Bjerhammar-sphere
The functions Ynm(P) are orthogonal basefunctions in a Hilbert
space with an isotropic innerproduct, harmonic down to a
so-called Bjerhammar-sphere
totally enclosed in the Earth.
T will not necessarily be an
element of such a space, but may be approximated arbitrarily
well with such functions. In spherical approximation the
ellipsoid is replaced by a sphere with radius 6371 km.
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Reproducing Kernel

K (P, Q) 

n 2



n 2
a 
 n 
 rr'
2
(2n  1)a 2  n
n

m  n
Ynm (P) Ynm (Q)
n 1
Pn (cos  )
r
P
ψ
r’
Q
where ψ is the spherical distance between P and Q,
Pn the Legendre polynomials and σn are positive
constants, the (potential) degree-variances.
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Inner product, Reproducing property
1

2
(2n  1)a  n


Ynn (P), Ynn (P)
Ynn
2
T ( P)  (T (Q), K ( P, Q)) 


n
 
n  2 m  n


GM Cnm Ynm (Q),
n  2 m  n

i2
(2i  1)a 2  i

j  i

n
 


i
GM Cnm Ynm ( P)

i2
Yij ( P)Yij (Q)

i
(2i  1)a 2  i
  Y (Q)Y (Q) 
j  i
ij
ij
n
 
n  2 m  n
GM Cnm Ynm ( P)  T ( P)
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Closed expression – no summation to

• the degree-variances are selected equal to simple
polynomial functions in the degree n multiplied by
exponential expressions like qn, where q < 1, then
K(P,Q) given by a closed expression. Example:
2 n 1
 RB 
 n   2   qn 1
R 

K (P, Q) 

n 0
R
2
B
R 
 
R 
2
B
2
n 1
R 
 
 rr' 
2
n 1
Pn (cos  ) 
1
R 4B  (rr') 2  2rr' R 2B
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Hilbert Space with Reproducing Kernel
• Everything like in an n-dimensional vector
space.
GM
• COORDINATES: a 2 Cnn
• ANGLES  between two
f , g
cos(  ) 
gf
• functions, f, g
• PROJECTION f ON g:
• IDENTITY MAPPING:
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g
 f , g
g
T(P)  T(Q), K(P, Q)
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Data and Model
In a (RKHS) approximations T from data for which the
associated linear functionals are bounded.
• The relationship between the data and T are expressed
through functionals Li,
yi = Li (T) + ei + A  X
T
i
yi is the i'th data element,
Li the functional, ei the error,
Ai a vector of dimension k and
X a vector of parameters also of dimension k.
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Data types
GEOCOL codes:
11
12
13
16
17
• Also gravity gradients,
• along-track or area
mean values.
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Test data
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Linear Functionals, spherical approximation
T
g = r
T 2T
g = r r
1 T
 =r 
1
T
=r cos(  ) 
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Best approximation: projection.
Ti pre-selected base functions:
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Collocation
LSC tell which functions to
select if we also require that
approximation and
observations agree and
how to find projection !
Suppose data error-free:
We want solution to agree with
data
We want smooth variation
between data
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Projection
Approximation to T using error-free data is obtained
using that the observations are given by, Li(T) = yi
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LSC - mathematical
• The "optimal" solution is the projection on the ndimensional sub-space spanned by the so-called
representers of the linear functionals, Li(K(P,Q)) =
K(Li,Q).
• The projection is the intersection between the
subspace and the subspace which consist of functions which agree exactly with the observations,
Li(g)=yi.
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Collocation solution in Hilbert Space
~
T (Q) 
n

bi K ( Li , Q)
i 1
Normal Equations
{bi } = {K Li , L j  } { y j }
-1
Predictions:
n
L(Tˆ) =  bi K( Li , L)
i=1
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Statistical Collocation Solution
We want solution with smallest “error” for all
configurations of points which by a rotation around the
center of the Earth can be obtained from the original
data. And agrees with noise-free data.
~
Li (T )  Li (T )
We want solution to be linear in the observations
~ ( P) 
T
n

i 1
n
 i yi 

i 1
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 i L i (T)
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Mean-square error - globally
~ (P)  T(P) 
T
n
  L (T)  T(P)
i 1

1
2 
8  

/2
 / 2

2
0
i
i
n
(T(P)    i L i (T))
2
i 1
d PPi cos( )d d
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Global Covariances:
CPi  COV(T(P), L i (T)) 
1
2  T( P) L i ( T) cos  d d d
8
C0 (T(P))  COV(T(P), T(P)) 
1
2
T
(
P
)
cos  d d d
2 
8
Cij  COV(L i (T), L j (T)) 
1
2  L j ( T) L i ( T) cos  d d d
8
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Covariance – series development
COV(P, Q)  COV(T(P), T(Q) 


i 2
2 i 1
R 
 i   Pi (cos  ),
 rr'
2
 GM 
i   
Cij  , DEGREE  VARIANCES
j i R
i
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Collocation Solution
~
T ( P) 
n

i 1
bi COV (T ( P), Li (T ))
b   COV (( L (T ), L (T ))  y 
1
i
i
j
i

   COV (( L (T ), L (T ) COV (T ( P), L (T ))
1
j
i
j
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Noise
• If the data contain noise, then the elements σij of the variancecovariance matrix of the noise-vector is added to K(Li,Lj) =
COV(Li(T),Lj(T)).
• The solution will then both minimalize the square of the norm of T
and the noise variance.
• If the noise is zero, the solution will agree exactly with the
observations.
• This is the reason for the name collocation.
• BUT THE METHOD IS ONLY GIVING THE MINIMUM
LEAST-SQUARES ERROR IN A GLOBAL SENSE.
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Minimalisation of mean-square error
The reproducing kernel must be selected equal to the
empirical covariance function, COV(P,Q).
This function is equal to a reproducing kernel with the
degree-variances equal to
 
2
n

m _ n
2
 GM  2

 Cnm
 a 
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Covariance Propagation
• The covariances are computed using the "law" of
covariance propagation, i.e.
• COV(Li,Lj) = Li(Lj(COV(P,Q))),
• where COV(P,Q) is the basic "potential"
covariance function.
• COV(P,Q) is an isotropic reproducing kernel
harmonic for either P or Q fixed.
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Covariance of gravity anomalies
ev P (T)  T(P)
T
2
 g(P)  
 ev P (T)
P
r
r
Appy the functionals on
K(P,Q)=COV(P,Q)

  2
COV( g(P), g(Q)) =  - - ev P  

 r r
 
  2
- evQ    i
 i=2
 r  r '
i+1
 R2 
  Pi cos 
 r r 
i -1  R 

 
=  - - ev   
  P cos 

 r
r '  r r 
(i -1 )  R 
= 
  P ( cos )
r r  r r 
I
2
i
P
i+1
i
i=2
2

i
2
i+1
i
i=2
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Evaluation of covariances
The quantities COV(L,L), COV(L,Li) and
COV(Li,Lj) may all be evaluated by the sequence
of subroutines COVAX, COVBX and COVCX
which form a part of the programs GEOCOL and
COVFIT.
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Remove-restore (I).
If we want to compute height-anomalies from
gravity anomalies, we need a global data
distribution.
If we work in a local area, the information outside
the area may be represented by a spherical
harmonic model. If we subtract the contribution
from such a model, we have to a certain extend
taken data outside the area into account.
(The contribution to the height anomalies must later be
restored=added).
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Change of Covariance Function
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Homogenizing the data
• minimum mean square error in a very specific
sense:
• as the mean over all data-configurations which by
a rotation of the Earth's center may be mapped into
each other.
• Locally, we must make all areas of the Earth look
alike.
• This is done by removing as much as we know,
and later adding it. We obtain a field which is
statistically more homogeneous.
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Homogenizing (II)
• 1.We remove the contribution Ts from a known spherical
harmonic expansion like the OSU91A field, EGM96 or a
GRACE model complete to degree N=360
• 2. We remove the effect of the local topography, TM, using
Residual Terrain Modelling (RTM): Earths total mass not
changed,
• but center of mass may have changed !!!
• We will then be left with a residual field, with a
smoothness in terms of standard deviation of gravity
anomalies between 50 % and 25 % less than the original
standard deviation.
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Residual quantities
yir = yi - Li ( T s ) - Li ( T M )
= Li (T) - Li ( T s ) - Li ( T M ) + ei + ATi  X
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Exercise 1.
• Compute residual gravity anomalies and deflections of the
vertical using the OSU91A spherical harmonic expansion
and the New Mexico DTM, cf. Table 1. The free-air
gravity anomalies are shown in
http://cct.gfy.ku.dk/geoidschool/nmfa.pdf
• The program GEOCOL may be used to subtract the
contribution from OSU91A.
• Job-files: http://cct.gfy.ku.dk/geoidschool/jobosu91.nmfa
• http://cct.gfy.ku.dk/geoidschool/jobosu91.nmdfv
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Output-files
Output from run:
http://cct.gfy.ku.dk/geoidschool/appendix2.txt
OSU91: http://cct.gfy.ku.dk/geoidschool/osu91a1f
Differences:
http://cct.gfy.ku.dk/geoidschool/nmfa.osu91
http://cct.gfy.ku.dk/geoidschool/nmdfv.osu91
Difference map:
http://cct.gfy.ku.dk/geoidschool/nmfaosu91.pdf
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Residual topography removal
The RTM contribution may be computed and
subtracted using the program tc1.
– First a reference terrain model must be constructed
using the program TCGRID with the file nmdtm as
basis, http://cct.gfy.ku.dk/geoidschool/nmdtm
– A jobfile to run tc1
– http://cct.gfy.ku.dk/geoidschool/jobtc.nmfa
– The result should be stored in files with names nmfa.rd
and nmdfv.rd, respectively.
– The residual gravity anomalies
– http://cct.gfy.ku.dk/geoidschool/nmfard.pdf
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Smoothing or Homogenisation
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Consequences for the statistical model.
• The degree-variances will be changed up to the
maximal degree, N, sometimes up to a smaller
value, if the series is not agreeing well with the
local data (i.e. if no data in the area were used
when the series were determined).
• The first of N new degree-variances will depend
on the error of the coefficients of the series. We
will here suppose that the degree-variances at least
are proportional to the so-called error-degreevariances,
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Error-degree-variances
E
=


n
n
n
2
=

  nm ,
E
n
m=-n
where  nm is the standard - deviation of GM/a  C nm
The scaling factor α must therefore be determined
from the data (in the program COVFIT, see later).
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Covariance function estimation and representation.
The covariance function to be used in LSC is equal to
COV(P,Q) =
2 /2 2
1
T(P)  T(Q) d cos  d d



8

• where α is the azimuth between P and Q and φ, λ are
the coordinates of P.
• This is a global expression, and that it will only
dependent on the radial distances r, r' of P and Q and
of the spherical distance ψ between the points.
2
0 - /2 0
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Global-local evaluation
• In practice it must be evaluated in a local area by
taking a sum of products of the data grouped
according to an interval i of spherical distance,
 i - /2  < i + /2
• Δψ is the interval length (also denoted the
sampling interval size).
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Covariance
• In spherical approximation we have already
derived
•
COV( g(P), g(Q))
R 
 n - 1
= 
 n  
 r r 
n=2  R 

2
2
n+2
Pi ( cos )
• where R is the mean radius of the Earth.
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Exercise 2.
• Compute the empirical gravity anomaly
covariance function using the program EMPCOV
for the New Mexico area both for the anomalies
minus OSU91A and for the anomalies from which
also RTM-effects have been subtracted (input files
nmfa.osu91 and nmfa.rd).
• A sample input file to EMPCOV is called
http://cct.gfy.ku.dk/geoidschool/empcov.nmfa, .
• A sample run is shown in Appendix 3. The
estimated covariances are shown in Fig. 5.
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Empirical Covariances
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Degree-variances
We see here, that if we can find the gravity anomaly
degree-variances, we also can find the potential
degree variances.
However, we also see that we need to determine
infinitely many quantities in order to find the
covariance function
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Model-degree-variances
• Use a degree-variance model, i.e. a functional dependence
between the degree and the degree-variances.
• In COVFIT, three different models (1, 2 and 3) may be
used. The main difference is related to whether the
(potential) degree-variances go to zero like n-2, n-3 or n-4.
The best model is of the type 2,
2
•
A
 RB 
 
 n=
(n - 1)(n - 2)(n - B)  R 
• where RB is the radius of the Bjerhammar-sphere, A is a
constant in units of (m/s)2, B an integer.
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COVFIT
• The actual modelling of the empirically
determined values is done using the program
COVFIT. The factors a, A and RB need to be determined (the first index N+1 must be fixed).
•
The program makes an iterative non-linear adjustment for the Bjerhammar-sphere radius, and linear
for the two other quantities
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Divergence ?
Unfortunately the iteration may diverge (e.g. result
in a Bjerhammar-sphere radius larger than R).
• This will normally occur, if the data has a very
inhomogeneous statistical character.
• Therefore simple histograms are always produced
together with the covariances (in EMPCOV) in
order to check that the data distribution is
reasonably symmetric, if not normal.
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Exercise 3.
Compute using COVFIT an analytic representation
for the covariance function.
An example of an input file is found in
http://cct.gfy.ku.dk/geoudschool/covfit.nmfa, . An
example of a run is shown in Appendix 3. Gravity
error-degree-variances for the OSU91A
coefficients are found in the file edgv.osu91.
• The estimated and the fitted covariance values are
shown above.
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Table of model-covariances
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LSC geoid determination from residual data.
• We now have all the tools available for using
LSC: residual data and a covariance model.
•
•
•
•
1.establish the normal equations,
2.solve the equations, and
3. compute predictions and error estimates.
This may be done using GEOCOL.
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Equations
• However, as realized from eq. (8) we have to solve a
system of equations as large as the number of
observations. GEOCOL has been used to handle 50000
observations simultaneously.
• This is one of the key problems with using the LSC
method. The problem may be reduced by using means
values of data in the border area.
• Globally gridded data can be used (sphgric)
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Necessary data density (d)
• Function of correlation length of the covariance function.
• We want to determine geoid height differences with an
error of 10 cm over 100 km. This corresponds to an error
in deflections of the vertical of 0.2".
• This is equivalent to that we must be able to interpolate
gravity anomalies with2
2
(d
0.3
/
)



C
0
e
1
d
a mean error of 1.2
mgal. The
where ed is the st.dev. ,
rule-of-thumb is
C 0 the gravity variance,
 1 the correlatio n distance
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Exercise 5. Data density.
• Use the residual gravity variance C0, and the
correlation distance determined in exercise 3 for
the determination of the needed data spacing.
• Then use the program SELECT for the selection
of points as close a possible to the nodes of a grid
having the required data spacing, and which
covers the area of interest.
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Exercise 5. Data selection.
• The area covered should be larger than the area in
which the geoid is to be computed. Data in a
distance at least equal to the distance for which
gravity and geoid becomes less than 10 %
correlated, cf. the result of exercise 3.
• Denote this file nmfa.rd1.
• When data have been selected (as described in
exercise 5) it is recommended to prepare a contour
plot of the data. This will show whether the data
should contain any gross-errors. LSC may also be
used for the detection of gross-errors.
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Exercise 5.GEOCOL INPUT.
An input file for the program GEOCOL must then
be prepared, or the program may be run
interactively.
In order to compute height-anomalies at terrain
altitude, a file with points consisting of number,
latitude, longitude and altitude must be prepared.
This may be prepared using the program GEOIP,
and input from a digital terrain model (nmdtm).
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Exercise 6.
• Prepare a file named nm.h covering the area
bounded by 33.0o and 34.0o in latitude and -107.0o
and -106.0o in longitude consisting of sequence
number, latitude, longitude and height given in a
grid with 0.1 degree spacing.
• Use the program GEOIP with input from nmdtm.
This will produce a grid-file. This must be
converted to a standard point data file (named
nmh2) using the program GLIST.
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GEOCOL INPUT/SPECIFICATIONS.
• the coordinate system used (GRS80),
• the spherical harmonic expansion subtracted (and
later to be added),
• the constants defining the covariance model and
contingently its tabulation
• the input data files (nmfa.rd or nmfa.rd1 if a
selected subset is used)
• the files containing the points in which the predictions should be made (nm.h2).
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GEOCOL OPTIONS
• Several options must be selected such as whether
error-estimates should be computed or whether we
want statistics to be output.
• produce a so-called restart file. This file is an
ASCII-file which contains input to GEOCOL
which enables the calculation of predictions only.
But it has the advantage that it may be used on
different computers.
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Exercise 7.
• Run the program GEOCOL (geocol16) with the
selected gravity data for the prediction of geoid
heights and their errors in the points given by
nm.h2.
• Output to a file named nm.geoid. Predict also
residual deflections of the vertical (nmdfv.rd) and
compare with the observed quantities.
• A model input file is found in jobnmlsc
• An example of a run where all data in a sub-area
are used is found in
http://cct.gfy.ku.dk/geoidschool/appendix5.txt .
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Exercise 7. RESTORE.
When the LSC-solution has been made, the RTM
contribution to the geoid must be determined.
Use tc1 with the file nm.h defining the points of
computation.
The LSC determined residual geoid heights and the
associated error-estimates are shown in
http://cct.gfy.ku.dk/geoidschool/nmgeoid.pdf
http://cct.gfy.ku.dk/geoidschool/nmgeoidh.pdf .
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Exercise 8.
• Compute the RTM contribution to the geoid using tc1 and
add the contribution to the output file from exercise 7,
nm.geoid.
• If mean gravity anomalies, deflections or GPS/levelling
determined geoid-heights were to be used, they could
easily have been added to the data.
• It would not be necessary to recalculate the full set of
normal-equations.
• Only the columns related to the new data need to be added.
Likewise, an obtained solution may be used to calculate
such quantities and their error-estimates.
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Exercise 9.
• Compute a new solution with the same observations as in
exercise 7, but add as observation one of the predicted
residual geoid heights. Define the error to be 0.01 m.
• Recalculate the geoid heights and the error-estimates.
• Use the possibility for re-using the Cholesky-reduced
normal-equations generated in exercise 7.
• Verify that the error-estimates, which now are equivalent
to error-estimates of geoid height differences, have a
magnitude smaller than the one specified in exercise 5.
(Error-estimates corresponding to one observed geoid
height are shown in
http://www.gfy.ku.dk/~cct/geoidschool/nmgeoidf.
pdf ).
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Exercise 9..
• The use of deflections and geoid heights (e.g.
from satellite altimetry) may require that
parameters such as datum shifts and bias/tilts are
determined. These possibilities are also included
in GEOCOL
• See next lecture.
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Conclusion (I)
• We have now went through all the steps from data
to predicted geoid heights.
• The exercises describes the use of gravity data
only, but observed mean gravity anomalies,
• GPS/levelling derived height anomalies as well as
deflections could have been used as well.
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Conclusion (II)
• The difficult steps in the application of LSC is the
estimation of the covariance function and subsequent
selection of an analytic representation.
• The flexibility of the method is very useful in many
circumstances, and is one of the reasons why the method
has been applied in many countries.
• If the reference spherical harmonic expansion is of good
quality, only a limited amount of data outside the area of
interest are needed in order to obtain a good solution.
•
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Conclusion (III)
• But if this is not the case, data from a large
border-area must be used so that a vast
computational effort is needed to obtain a
solution.
• This may make it impossible to apply the method.
• A way out is then to use the method only for the
determination of gridded values, which then may
be used with Fourier transform techniques or Fast
Collocation.
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