Problems associated with using an altimetry sampling function approach to the geoid are examined. They include: (1) conventent mathematical representation of short-wavelength (eventually approximately 1 deg) features of the geoid or geopotential, (2) utilization of detailed data from only part of the globe (i.e., the oceans) (3) application of appropriate formalism to relate the sea-level equipotential below the atmospheric mass to the external potential above the atmosphere, (4) mathematical applicability of an adopted geopotential representation on the surface of the physical geoid

Giacaglia, G. E. O.

Lundquist, C. A.

English

NASA Technical Reports Server

ii
"i
USE OF ALTIMETRY DATAt IN A SAMPLING-
FUNCTION APPROACH TO THE GEOID
C. A. Lundquist
Smithsonian Astrophysical Observatory, Cambridge, Massachusetts
and
G. E. O. Giacaglia T
University of Texas, Austin, Texas
.
The planned operation of satellite-to-ocean altimeters will produce meas-
urements that require mastery of particular data-analysis problems for the full
utilization of the information in these measurements. Under the premises that
the first altimeters will have an accuracy of ~ 1 m and that at this scale the ocean
profile can be identified with an equipotential surface, the following problems are
among those that must be examined:
1. Convenient mathematical representation of short-wavelength (eventually
~ 1 °) features of the geoid or geopotential.
2. Utilization of detailed data from only part of the globe (i. e., the oceans).
3. Application of appropriate formalism to relate the sea-level equipoten-
tial below the atmospheric mass to the external potential above the atmosphere.
4. Mathematical applicability of an adopted geopotential representation on
the surface of the physical geoid.
These topics are not independent, of course.
This research was supported in part by grant NGR 09-015-002 from the National
Aeronautics and Space Administration.
_On leave from the University of S_o Paulo, S_o Paulo, Brazil. Partially supported
by ONR Contract N00014-67-A-0126-0013.
110-026
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iThe concept of using a sampling-function representation of the geoid and
geopotential emerged from efforts to prepare for some of these problems, and
the evolution of this concept can be followed in other papers (Lundquist and
Giacaglia, 1969; 1971a, b; Giacaglia and Lundquist, 1971). The objective here
is rather to review the current status of the sampling-function representation
as a partial answer to the analysis problems posed by altimetry data.
With respect to the first problem - a convenient representation of short-
wavel.ength features -the coefficients in an expansion in sampling functions are
essentially tabular values of the geoid radius or potential at a grid of sampling
points on a sphere or similar reference surface. The grid can be scaled as
finely as desired. The sampling-function representation through some degree is
equivalent to a spherical-harmonic expansion tlu:ough the same degree, and the
transformation from sampling functions to spherical harmonics and its inverse
are expressed in analytical form (Lundquist and Giacaglia, 197 lb). Therefore,
no need arises to invert large matrices numerically, and this aspect of the
altimetry problem is resolved.
In an oversimplified scenario for the treatment of altimeter data, each alti-
tude measurement from a determinable position in orbit implies a geocentric
radius to the ocean surface. All these measurements of radii in the neighbor-
hood of a sampling point can be accumulated and averaged appropriately to give
the radius at the point. This radius value is immediately the coefficient of the
corresponding sampling function in the geoid representation. If the equivalent
spherical-harmonic expansion is desired, this is obtainable by applying the
analytically defined transformation.
Some recent progress toward implementing these calculations has been the
preparation at the University of Texas of computer algorithms to evaluate the
necessary analytical formulas for fairly high degree. Even though simpler than
some other approaches, the calculations involved are extensive, owing to the
great detail of the desired representation. In the interest of computer efficiency,
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:!
the formulation of the analytical expressions and the computer algoritluns have
progressed through several steps of refinement.
Degree 36 has been selected for exploratory investigations, although a still
higher degree might be more illuminating. In this case, features with wave-
lengths as short as 5 ° can be represented. For an expansion through degree 36,
there are (36 + 1) 2 = 1369 terms in either a sampling-function or a spherical-
harmonic expansion. The transformation matrices relating the equivalent forms
have nearly two million elements.
As a trial application using the sampling points for degree 36, geocentric
radii were calculated to an equipotential surface derived by use of the
Smithsonian harmonic coefficients presented at the 1971 IUGG meeting
(Gaposchkin, Kozai, Veis, and Weiffenbach, 1971). This calculation at the
University of Texas followed the procedure discussed by Lm_dquist and Giacaglia
(197 la}. Als% geocentric radii were calculated (Girnius, 1971) for 45 sampling
points in the North American Datum, by use of the Army Map Service 1967 Map
of Geoid Contours in North America from Astrogeodetic Deflections (Fischer, 1966).
Figure 1 shows the 45 sampling points. The geoid heights were transformed to
geodentric radii in 1969 Smithsonian Standard Earth (II) coordinates by using the
Lambeck (1971) parameters, assuming the Smithsonian and North American Datum
axes are parallel.
The radius values from the astrogeodetic geoid could contain somewhat shorter
wavelength information than the values from the Standard Earth. To generate a
sampling-function representation corresponding to the astrogeodetic geoid in
North America, it is only necessary to replace the Smithsonian values with those
from the geoid map for the sampling points in North America. This has been done.
If one wants the equivalent spherical-harmonic representation, the analytically
defined linear transformation can be applied.
Because a very similar operation is envisioned when satellite-to-ocean alti-
tudes are available, a study of the properties of this modified geoid representation
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k.
!:i
should indicate the utility of this method. Such a study is in progress. P_trtial
answers to both problems 1 and 2 are expected as a result of the trial application
to the North American geoid, since this test involves features of both problems.
For problem 1, a crucial aspect is the ability of the sampling-function repre-
sentation to reproduce short-wavelength features in North America. For problem
2_ the crucial question is whether extraneous short-wave detail is introduced with
significant amplitude for the geoid outside North America. The desired result
should be a geoid in North America resembling the astrogeodetic contours in its
5 ° and longer wavelength features, with the properties of the satellite-determined
field elsewhere. Also, the corresponding geopotential should have essentially the
Smithsonian coefficients for the lower degree and order spherical harmonics. An
iterative scheme may be necessary to achieve these properties.
The discussion and procedures above have been based on the implicit assump-
tion that the geopotential derived from satellite observations is also applicable at
the surface of the earth. WMle this is an acceptable simplification for exploratory
studies, it certainly must be reconsidered for accurate treatment of actual altitude
measurements. Problems 3 and 4 recognize the need to proceed with caution.
The mass of'the atmosphere is given by Verniani i1966) as 8. 594 × 10 -7 of the
mass of the earth. Clearly this mass contributes differently to the gravitational
field at satellite altitudes than it does at sea level. The first step to accommodate
this situation would seem to be a decomposition of the external potential into a
major portion due to the mass of the solid earth and oceans and a minor portion
due to the mass of the atmosphere.
The leading term in the usual spherical-harmonic expansion is proportional
to total mass, so that its coefficient can be decomposed into two fractions - res-
pcctivcly, 0.999, 999, 140, 6 and 0. 000, 000, 859, 4 of the total. Such an adjust-
mcnt was made by Veis (1967) in a determination of the equatorial radius and
gravity of the earth. This effect was noted also by Rapp (1970) in a discussion of
.............. _:i:i__ . _::• ..... :: ............. i__;,:i •_:i_: •.:..: : :/_i:_::: :_: .......... ii
methods for the computation of geoid undulations from potential coefficients and
•:_ by others in other contexts (Ecker, 1968; Ecker and Mittermayer, 1969).
'.j
.:!
Since the atmosphere is constrained to a nearly ellipsoidal lower boundary
by the shape of the solid earth and oceans, its mass must make a contribution to
the total J2 of the earth. A first crude estimate of the size of this contribution
is obtained by considering the total mass of the atmosphere concentrated in a uni-
form ellipsoidal shell with the same semimajor axes as the earth. This crude
estimate gives J2 (atmosphere} = 0.002 X 10 -6 as compared with the Kozai value
J2 = (1082. 637 + 0. 001) X 10 -6 for the total earth system. Thus, the contribution
of the solid earth and of the oceans would be J2 (solid earth and oceans} =
1082. 635 X 10 -6. This very small change would not seem to be important until
geoid accuracies in the centimeters are obtained.
On the other hand, Kozai reports an annual variation of amplitude
5J 2 = 0. 0013 X 10-6_ presumably due to mass displacements somewhere in the
earth system (Kozai, 1970). A more accurate calculation of the atmospheric con-
tribution to J2 would be instructive_ to improve the crude estimate above. Kelly
(1971) has assembled the atmospheric models and formulas for such a calculation.
In principle, there is a further complication associated with the atmosphere -
namely_ the gravitational field at sea level due to the nearly elliptical atmospheric
shell above. This contribution should be added back into the potential after the
external atmospheric contribution has been subtracted from satellite information to
isolate the field due to the solid earth and oceans. However, this internal field of
the atmosphere is probably even less important than correction of the J2 value.
The fourth problem, the mathematical applicability of an adopted geopotential
representation at sea level, is a perplexing one in potential theory (see, for
example, Hotine, 1969; Madden, 1971). It has been argued that the convergence
uncertainties expected with a spherical-harmonic expansion could be largely alle-
viated by the use of ellipsoidal harmonics (see, for example, Madden, 1968;
Walter, 1971), presumably because the ellipsoidal functions can better conform
to the shape of the earth.
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i ..... "_"_'-"" .... • -_" "_"_'-'_i!
The sampling functions can be defined on an ellipse about as easily as on a
reference sphere, and if the elliptical formulation is used, it would seem that
they should accrue the same benefits as ellipsoidal harmonics. Still further,
the sampling functions can also be defined on a surface conforming still more
closely to the geoid. It is an open question whether this would still further alle-
viate the convergence uncertainty.
In summary, although many questions remain to be answered, a sampling-
function representation of the geoid still promises to be a useful tool in utilizing
satellite-to-ocean altitudes.
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REFERENCES
ECKER, E. •
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1969. Gravity corrections for the influence of the atmosphere. Boil. Geofisica
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Use of altimetry data in a sampling-function approach to the geoid

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