Geoid determination by

1

• Geoid determination by
• least-squares collocation using
• GRAVSOFT

C.C.Tscherning, University of Copenhagen,
2005-01-28
1
2
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

C.C.Tscherning, University of Copenhagen,
2005-01-28
2
3
Quasi-geoid

• Important
• the term geoid the quasi-geoid,
• i.e. the height anomaly evaluated at the surface
  of the Earth.

C.C.Tscherning, University of Copenhagen,
2005-01-28
3
4
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
4
5
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
5
6
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
6
7
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).

C.C.Tscherning, University of Copenhagen,
2005-01-28
7
8
Covariance

• Determine at statistical model (a covariance
  function) for the residual data in the region in
  question.
• EMPCOV, COVFIT

C.C.Tscherning, University of Copenhagen,
2005-01-28
8
9
Select

• Make a homogeneous selection of the data to be
  used for geoid determination using
  rule-of-thumbs for the required data density,
  SELECT
• If many data select those with the smallest error
  XSelection of points O closest to the middle. 6
  points selected

X
o
o
o
x
X
o
x
o
o
x
x
C.C.Tscherning, University of Copenhagen,
2005-01-28
9
10
Errors

• check for gross-errors (make histograms and
  contour map of data), GEOCOL
• verify error estimates of data, GEOCOL.

C.C.Tscherning, University of Copenhagen,
2005-01-28
10
11
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
11
12
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 !

C.C.Tscherning, University of Copenhagen,
2005-01-28
12
13
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
13
14
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
• GM is the product of the gravitational constant
  and the mass of the Earth and the fuly
  normalized spherical harmonic coefficients.

C.C.Tscherning, University of Copenhagen,
2005-01-28
14
15
Coordinates used.

• GEOCOL accepts geocentric, geodetic and Cartesian
  (X,Y,Z) coordinates but output only in geodetic.

C.C.Tscherning, University of Copenhagen,
2005-01-28
15
16
Solid spherical harmonics.

• where a is the semi-major axis and Pnm the
  Legendre functions.
• We have orthogonality

C.C.Tscherning, University of Copenhagen,
2005-01-28
16
17
Bjerhammar-sphere

• The functions Ynm(P) are ortho-
• gonal basefunctions in a Hilbert
• space with an isotropic inner-
• product, 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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
17
18
Reproducing Kernel

• where ? is the spherical distance between P and
  Q, Pn the Legendre polynomials and sn are
  positive constants, the (potential)
  degree-variances.

P
r
Q
?
r
C.C.Tscherning, University of Copenhagen,
2005-01-28
18
19
Inner product, Reproducing property
C.C.Tscherning, University of Copenhagen,
2005-01-28
19
20
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 lt 1,
  then K(P,Q) given by a closed expression. Example

C.C.Tscherning, University of Copenhagen,
2005-01-28
20
21
Hilbert Space with Reproducing Kernel

• Everything like in an n-dimensional vector space.
• COORDINATES
• ANGLES ? between two
• functions, f, g
• PROJECTION f ON g
• IDENTITY MAPPING

C.C.Tscherning, University of Copenhagen,
2005-01-28
21
22
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 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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
22
23
Data types

• GEOCOL codes
• 11
• 12
• 13
• 16
• 17
• Also gravity gradients,
• along-track or area mean values.

C.C.Tscherning, University of Copenhagen,
2005-01-28
23
24
Test data
C.C.Tscherning, University of Copenhagen,
2005-01-28
24
25
Linear Functionals, spherical approximation
C.C.Tscherning, University of Copenhagen,
2005-01-28
25
26
Best approximation projection.

• Ti pre-selected base functions

C.C.Tscherning, University of Copenhagen,
2005-01-28
26
27
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

C.C.Tscherning, University of Copenhagen,
2005-01-28
27
28
Projection

• Approximation to T using error-free data is
  obtained using that the observations are given
  by, Li(T) yi

C.C.Tscherning, University of Copenhagen,
2005-01-28
28
29
LSC - mathematical

• The "optimal" solution is the projection on the
  n-dimensional 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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
29
30
Collocation solution in Hilbert Space

• Normal Equations
• Predictions

C.C.Tscherning, University of Copenhagen,
2005-01-28
30
31
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.
• We want solution to be linear in the observations

C.C.Tscherning, University of Copenhagen,
2005-01-28
31
32
Mean-square error - globally
C.C.Tscherning, University of Copenhagen,
2005-01-28
32
33
Global Covariances
C.C.Tscherning, University of Copenhagen,
2005-01-28
33
34
Covariance series development
C.C.Tscherning, University of Copenhagen,
2005-01-28
34
35
Collocation Solution
C.C.Tscherning, University of Copenhagen,
2005-01-28
35
36
Noise

• If the data contain noise, then the elements sij
  of the variance-covariance 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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
36
37
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

C.C.Tscherning, University of Copenhagen,
2005-01-28
37
38
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
38
39
Covariance of gravity anomalies

• Appy the functionals on
• K(P,Q)COV(P,Q)

C.C.Tscherning, University of Copenhagen,
2005-01-28
39
40
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
40
41
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 restoredadded).

C.C.Tscherning, University of Copenhagen,
2005-01-28
41
42
Change of Covariance Function
C.C.Tscherning, University of Copenhagen,
2005-01-28
42
43
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
43
44
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
  N360
• 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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
44
45
Residual quantities
C.C.Tscherning, University of Copenhagen,
2005-01-28
45
46
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/jobos
  u91.nmfa
• http//cct.gfy.ku.dk/geoidschool/jobosu91.nmdfv

C.C.Tscherning, University of Copenhagen,
2005-01-28
46
47
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

C.C.Tscherning, University of Copenhagen,
2005-01-28
47
48
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/geoidsch
  ool/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

C.C.Tscherning, University of Copenhagen,
2005-01-28
48
49
Smoothing or Homogenisation
C.C.Tscherning, University of Copenhagen,
2005-01-28
49
50
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-degree-variances,

C.C.Tscherning, University of Copenhagen,
2005-01-28
50
51
Error-degree-variances

• The scaling factor a must therefore be determined
  from the data (in the program COVFIT, see later).

C.C.Tscherning, University of Copenhagen,
2005-01-28
51
52
Covariance function estimation and representation.

• The covariance function to be used in LSC is
  equal to
• where a is the azimuth between P and Q and f, ?
  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 ? between the
  points.

C.C.Tscherning, University of Copenhagen,
2005-01-28
52
53
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,
• ?? is the interval length (also denoted the
  sampling interval size).

C.C.Tscherning, University of Copenhagen,
2005-01-28
53
54
Covariance

• In spherical approximation we have already
  derived
• where R is the mean radius of the Earth.

C.C.Tscherning, University of Copenhagen,
2005-01-28
54
55
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
55
56
Empirical Covariances
C.C.Tscherning, University of Copenhagen,
2005-01-28
56
57
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

C.C.Tscherning, University of Copenhagen,
2005-01-28
57
58
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,
• where RB is the radius of the Bjerhammar-sphere,
  A is a constant in units of (m/s)2, B an integer.

C.C.Tscherning, University of Copenhagen,
2005-01-28
58
59
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 N1 must be fixed).
• The program makes an iterative non-linear
  adjustment for the Bjerhammar-sphere radius, and
  linear for the two other quantities

C.C.Tscherning, University of Copenhagen,
2005-01-28
59
60
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
60
61
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
61
62
Table of model-covariances
C.C.Tscherning, University of Copenhagen,
2005-01-28
62
63
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
63
64
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)

C.C.Tscherning, University of Copenhagen,
2005-01-28
64
65
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 with
• a mean error of 1.2
• mgal. The
• rule-of-thumb is

C.C.Tscherning, University of Copenhagen,
2005-01-28
65
66
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
66
67
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
67
68
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).

C.C.Tscherning, University of Copenhagen,
2005-01-28
68
69
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
69
70
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).

C.C.Tscherning, University of Copenhagen,
2005-01-28
70
71
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
71
72
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/geoidsch
  ool/appendix5.txt .

C.C.Tscherning, University of Copenhagen,
2005-01-28
72
73
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 .

C.C.Tscherning, University of Copenhagen,
2005-01-28
73
74
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
74
75
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/nmg
  eoidf.pdf ).

C.C.Tscherning, University of Copenhagen,
2005-01-28
75
76
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
76
77
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
77
78
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
78
79
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.

C.C.Tscherning, University of Copenhagen,
2005-01-28
79