**Covering the GOCE mission polar data
gaps using gradients and ground gravity.**

** **

**C.C.Tscherning**

*Institut de Geomàtica,
Barcelona, Catalunya.*

**On leave from:**

**University of Copenhagen, Department of Geophysics,**

**Juliane Maries Vej 30, DK-2100 Copenhagen Oe., Denmark**

*e-mail: **cct@gfy.ku.dk*

* *

**Abstract**

Least-squares collocation error estimates of spherical
harmonic coefficients have been simulated using Gravity gradients (T_{zz }) in the area covered by GOCE and
either upward continued T_{zz} data or ground gravity anomaly data at
one or both poles. An isotropic covariance function using the EGM96 (statistically independent) error
degree variances to degree 180 and degree variances derived from Wenzels GPM98A
were used at higher degrees. Two kinds
of data distributions have been considered.
Equal-area distributions with a spacing of 1 degree and 2 deg. and
equal-angular distributions with spacing 2 deg., 1 deg. and 0.5 deg.

The error of near-zonal harmonics were typically 2 times
larger than the other harmonics when the poles were not covered. A slight
improvement was seen when T_{zz } or gravity data on one pole was used, and
uniform error estimates were obtained if both poles were covered, however for
the gravity data dependent on the supposed associated noise standard deviation
and height.

It was found in one example that if ground gravity
should be able to "fill the gaps" they should have a resolution twice
the one of the T_{zz} data. In the 2-degree experiment, 1 degree-mean
gravity anomaly data with a 0.2 mgal error were needed in order to "fill
the gap" for the coefficients of degree 28, for example.

**1. Introduction.**

** **

The
selected orbit of the GOCE mission creates gaps in the data coverage at the
poles. These gaps may however be covered using other data-types such as ground
or airborne gravity data.

How
much do we need to fill the gaps, i.e. have a uniform precision of the
estimated spherical harmonic coefficients ?

In
order to study the influence of adding such data, simulations have been
performed using least-square collocation (LSC), (Moritz, 1980) for the
computation of the error-estimates of
the spherical harmonic coefficients using different data-combinations. We
first describe how it is planned to use LSC to process GOCE data using data in
the points where they are observed and secondly using gridded data at satellite
altitude. Then we describe the results of using the two methods for simulation
studies. The first procedure has limitations with respect to how many data can
be treated simultaneously. The second have limitations associated with respect
to which data-types can be used, but very large data-sets can be used.

**2. Use of LSC for the processing of GOCE data.**

The
use of LSC for the processing of GOCE data is described in detail within the framework of the so-called
space-wise approach in the E2M report, (Albertella et al., 2000, Tscherning et
al., 2000a, b). Here the main features and some recent developments are summarized.

In
general LSC permit the use of all data-types, both satellite gravity
gradiometry (SGG) and satellite-to-satellite tracking (SST), data,
simultaneously. Also ground data may be added. However a system of equations as
large as the number of observations will have to be solved. Methods have been
found (Moreaux et al., 1999) which permit the handling of very large
datasets using sparse matrix
techniques. The present author does not doubt that at the time of the launch of
GOCE (2005) it will be possible to treat simultaneously all vertical gravity
gradient data simultaneously.
Unfortunately data are not strictly vertical gravity gradient data. The data
will be given in a satellite reference frame, and it will not be possible to
“rotate” the quantities into a reference frame having e.g. the radius-vector as
one of its axes. (This requires that all gravity gradient matrix components are
measured). At least if one tries to “rotate” new data errors will be
introduced.

Instead
regional gravity field approximations can be determined from any available
data-type. In 2005 it will certainly be
feasible to handle at least 200000 observations in one run. (41000 took 3 days
on a 500 MHz PC in 2000).

Suppose
we have 3 observations every 4 seconds (potential differences from SST, T_{zz}
, T_{xx} – T_{yy} from SGG). Then one regional solution will
cover a 20 degree x 20 degree area.

The
area covered by GOCE will have to be covered by equal-area blocks with e.g. 2.5
degree overlap. This means 24 blocks at Equator, about 190 globally. This is a
reasonable number of blocks to handle.

Each
regional solution will then be used in
the calculation of normal values such as along-track filtered T_{zz } values. (The prediction will take care of the transformation from
the satellite frame to the Earth-oriented, radius-vector frame). Such a process
should help in assuring that a minimum of
information loss will occur when normal point values are constructed.
LSC with sparse matrices can then be used to handle global data-sets of such
normal values.

Simultaneously
a gridding can be performed, so that a regular grid in longitude is
constructed. (It does not need to be regular in latitude, or have the same
heights for each parallel). These data can be handled using the new method of
Fast Collocation (Sanso’ and Tscherning, 2001). This method requires, besides
the longitudional gridding, that a uniform noise is used for each data-type at
each parallel. This is not unrealistic considering the uniform data noise expected for GOCE data and
the fact that the orbits converge towards the poles. The method takes advantage
of the repetitive structure of the normal equations. If N parallels are
used, systems of equations with
dimension N needs to solved. If
coefficients up to degree and order M needs to be estimated (and their error
estimates computed), M of these systems of equations have to be solved. The
value of M can maximally be 180/(grid spacing) for the zero order terms, but
the double for the maximal order coefficients. Note that the more general
method discussed in 3.1. formally is bound by this Nyquist limit.

For
both procedures it is possible to use data of a different kind at the poles.
For sparse collocation gravity data must be converted – by upward continuation
– to e.g. T_{zz } data at altitude. For the Fast Collocation
method no upward continuation is needed, but the gravity data must be gridded
in longitude and for each parallel associated with the same height.

**3. Simulations using least-squares collocation.**

** **

**3.1 Using “randomly” distributed data.**

Least-squares
collocation has been used to determine the error-estimates of (correction to)
spherical harmonic coefficients as described in (Tscherning, 2001). Initially a
1 degree approximate equal area data distribution using T_{zz} at 300
km altitude and a data noise of 0.005 EU was used. This corresponds to approximately 41000 “observations” with a normal-equation (upper triangular) matrix of size 6 GB. An isotropic covariance function having
degree variances derived from the EGM96 coefficient errors (Lemoine et al., 1997),
regarded as uncorrelated, was used. The degree variances above degree 180 were
put close to zero. ( One may discuss whether it is realistic to use an
isotropic covariance function. However, considering that CHAMP and GRACE will
be flying before GOCE, and having near-polar orbits, we should expect that the
near zonal harmonic coefficients will have errors similar to those of the other
coefficients.)

The
typical error pattern seen in other studies was also found, i.e. the
“near-zonal” harmonic coefficients (order 0, 1, 2, 3) had an error 50% times
larger than the other coefficients. (In experiments using a noise variance of
only 0.0005 EU the error was 10 times
larger).

It was
found that for the type of simulation presented here, a 2-degree coverage was
sufficient in order to gain insight into the problem. This correspond to about
10000 observations. The resulting
error-estimates are shown in Fig. 1.

The
results illustrated in this and in the following figures show
results for degree 28. This value is chosen quite arbitrarily, but the
results are representative for other degrees. Note, that at this degree the
standard deviation of the error of a single EGM96 coefficient is 4.3 x 10^{-9} (unitless).

Figure
1.

Ground
gravity data may be used to calculate upward continued gravity gradients at
satellite altitude. LSC was used to calculate the errors of upward continued
air-borne data with a data-distribution like the one obtained for Greenland
(see Brozena et al., 1997). As a covariance function the one derived from the
Greenland data was used It has a gravity anomaly standard deviation of 42 mgal
when subtracting the contribution from a high order reference field to degree
35.

Air-borne
gravity data measured with an error of 1 mgal was upward continued to T_{zz} values which resulted in an associated
estimated error of 0.006 EU, close to
the expected error of the gradiometer. (However, it should be noted that the
standard deviation of the signal (T_{zz } ) after subtraction of EGM96 to degree 180 is only 0.011 EU).

Two
simulations have then been made with these data, where first the North pole
and then the South Pole was filled, see Fig. 1. We see that in order to really
obtain a substantial reduction of the error of the “near” zonal coefficients,
data at both poles are needed.

Least-squares
collocation permit the combination of data of different types. Simulations were
therefore made using ground gravity upward continued to 20 km and 10 km
altitude. (This corresponds to 2 degree-equal area means and 1 degree equal
area means). The standard deviation of
these values as derived from typical modern air-borne gravity with a 1 mgal
error was found to be 0.6 mgal for gravity anomalies at 10 km altitude, and somewhat lower for the
20 km data.

The 20
km altitude gravity data gave a slight improvement. But it was first when the
10 km data was used that error-estimates similar to the ones obtained using
gravity gradients were obtained, see Fig. 2.

Data
error estimates of 0.5 and 0.2 mgal standard deviation were used, and it was
found that 0.2 mgal was needed.

Figure
2.

**3.2 Using gridded data.**

Grids
of data equidistantly located on each parallel with a density of 2 deg., 1 deg. and 0.5 deg. were used.
Obviously, the denser the data, the better the result, until a limit where it
is the data-noise which determines the error. Results similar to those obtained
using equal-area grids of T_{zz} (section 3.1) were obtained. However the coefficient errors were not
uniform but had a minimum for orders in between the maximal order and 0 order,
see Fig. 3, 4 and 5. Also in these experiments an uncorrelated data noise of 0.005 EU was used. An
isotropic

Figure
3.

Figure
4.

Figure
5.

covariance
function with degree-variances equal to the EGM96 error-degree-variances to
degree 180 and degree-variances from degree 181 to 1440 equal to the
degree-variances of GPM98A (Wenzel, 1998) was used.

Further
tests with gravity anomaly data with the same spacing as the gravity gradient data at altitudes 20, 10 and 5 km for the
polar areas (above 82 deg. and below –82 deg.) are planned.

**4. Conclusion:**

Gravity
data in the Arctic is available from airborne and sub-marine carried
gravimeters, (Tscherning et al., 2000)
and has error-estimates of between 5 and 10 mgal. Consequently they will not
improve the GOCE solution if data from CHAMP and GRACE are available. The data
distribution and quality is not sufficient
at present to obtain 0.006 EU for upward continued vertical gravity
gradients, but a further improvement of air-borne gravimetry is to be expected.
Otherwise we must use airborne gradiometry, which is also now an operational
procedure.

A
further improvement equivalent to a complete data coverage with gravity
gradients is possible only if both the Arctic and the Antarctic is covered
with good quality data.

**References:**

Albertella
A., F.Migliaccio, F.Sanso' and C.C.Tscherning: The space-wise approach -
Overall scientific data strategy. H.Suenkel (ED.) Eoetvos to mGal, Final
report, pp. 267-297, April 2000.

Brozena, J.: The Greenland Aerogeophysics Project.
Airborne Gravity, topographic and magnetic mapping of an entire continent. In:
O.Colombo (Ed.): From Mars to to Greenland. Proc. IAG Symp. G3, Vienna, Austria,
Aug. 1997, Springer Verlag, 1992.

Lemoine, F.G., D.Smith, R.Smith, L.Kunz, E.Pavlis,
N.Pavlis, S.Klosko, D.Chinn, M.Torrence, R.Williamson, C.Cox, K.Rachlin,
Y.Wang, S.Kenyon, R.Salman, R.Trimmer, R.Rapp and S.Nerem: The development of
the NASA GSFC and DMA joint geopotential model. Proc. Symp. on Gravity, Geoid
and Marine Geodesy, Sept. 30 - Oct. 5, 1996. The University of Tokyo, Tokyo,
1996.

Moritz, H.: Advanced Physical Geodesy. H.Wichmann
Verlag, Karlsruhe, 1980.

Sanso', F. and C.C.Tscherning: Fast
spherical collocation. Paper prepared for IAG2001, Budapest, Sept. 2001.

Tscherning, C.C., R.Forsberg, A.Albertella,
F.Migliaccio & F.Sanso': Space-wise approaches to gravity field
determination in Polar Areas. H.Suenkel (ED.) Eoetvos to mGal, Final report,
pp. 331-336, March 2000.

Tscherning,
C.C., G.Moreaux, A.Albertella, F.Migliaccio, F.Sanso' & D.Arabelos:
Detailled scientific data processing using the space-wise approach. H.Suenkel
(ED.) Eoetvos to mGal, Final report, pp. 299-304. March 2000.

Tscherning,
C.C., R.Forsberg, A.Albertella, F.Migliaccio & F.Sanso': Space-wise
approaches to gravity field determination in Polar Areas. H.Suenkel (ED.)
Eoetvos to mGal, Final report, pp. 331-336, March 2000.

Tscherning,
C.C.: Computation of spherical harmonic coefficients and their error estimates
using Least Squares Collocation. Accepted Journal of Geodesy, 2001.(Also in
E2M report).

Wenzel, H.G.: Ultra hochaufloesende
Kugelfunktionsmodelle GMP98A und GMP98B des Erdschwerefeldes. Proceedings
Geodaetische Woche, Kauserslautern, 1998.

File: polaregap3.doc, date: 2001-04-19