The ability of three high-order geopotential models (OSU81, GPM2 and OSU86E) to recover the gravity anomaly field (delta g) in the Australian region was tested. The region was divided into 2 x 2 deg blocks, and the mean and rms of the residual gravity (delta g measured - delta g modeled) was found to estimate the fit of the model to the point gravity data. The results showed that OSU81 and GPM2 performed similarly, recovering the delta g with a mean value of less than plus or minus 5 mGal in 63 and 70 percent of the blocks, respectively. However, both these models achieved a fit of worse that was plus or minus 13 mGal in 6 to 7 percent of cases. These were in areas either on or near the coast, or in the Central Australian region, inferring that for a precise geoid slope determination in these regions, a detailed analysis of delta g in region is needed. On the other hand, OSU86E produced a very good result, having a mean fit of less than plus or minus 5 mGal in 80 percent of the blocks, and worse than plus or minus 13 mGal in only 1 percent of cases. The rms values for this model were also improved over the other two models, indicating that for applications requiring highest precision, the preferred model is OSU86E

Holloway, R. D.

Kearsley, A. H. W.

English

NASA Technical Reports Server

GEOPOTENTIAL MODEI,S IN TilE AUSTRALIAN RE(;ION
,... w. N90-20519
R. D. IIollnway
School of Surveying, University of New South Wales, Kensington, Australia
Ahstract
We test the ability of three high-order geopotential models - OSU8I, GPM2 and OSU86E - to recover the
gravity anomaly field (Ag) in the Australian region. The region was divided into 2° x 2° blocks, and the mean and
rms of the residual gravity (Ag measured" Ag modelled) found to estimate the fit of the model to the point gravity
data. The results showed that OSUS1 and GPM2 performed similarly, recovering the Ag with a mean value of
< +_5 reGal in 63% and 70% of the blocks, respectively. However, both these models achieved a fit of worse that
+ 13 reGal in 6 to 7% of cases. These were in areas either on or near the coast, or in the Central Australian
region, inferring that for a precise geoid slope determination in these regions, a detailed analysis of Ag in region is
needed. On the other hand, OSU86E produced a very good result, having a mean fit of < + 5 mGal in 80% of the
blocks, and worse than + 13 mGal in only 1% of cases. The rms values for this model were also improved over
the other two models, indicating that for applications requiring highest precision, the preferred model is OSU86E.
1. Introduction
The growing interest in recovering orthometric heights from GPS has generated the need for models capable of
recovering geoid features to finer detail. The past decade has seen the development of a series of models of
increasing order, including two models taken to nmax = 180 (OSU79 and OSU81; Rapp, 1981), a model
summed to nmax = 200 (GPM2; Wenzel (1985)), and most recently, two models taken to nmax= 360, OSU86E
and OSU86F (Rapp, 1986). Geodesists have used these models extensively for geoid studies, mainly to
determine the geoidal long to medium wavelength features and to provide the reference model for terrestrially-
derived gravity anomalies used in, e.g., Stokes' integral, to find the short wavelength component of the geoid
signal.
The ability of GPS to determine eUipsoidal height differences (All) over lines whose orthometric height differences
(AH) are found by conventional levelling has provided, for the first time in history, high precision geometric
determinations of AN, the geoid height change over the baseline. That is, for small deflections of the vertical
AN= Ah- AH (1)
AN will have a precision equivalent to ¢_Ah, providing H is determined to 1st or 2nd order. This precision is
thought to be of the order of 2 to 4 ppm of the line length. Networks in which Ah and AH have been precisely
evaluated, therefore, provide valuable control data against which to compare gravimetric determinations of AN. We
have recendy performed a series of such comparisons, and have found that (Kearsley, 1988)
(i) the ability of a geopotential model to recover AN varies significantly, both with location and with nma x,
(ii) the mean fit of AN from the geopotential model to the control AN constrains the precision obtainable from a
full gravimetric solution
In the discussion which follows we test the fit of three models (OSU81, GPM2 and OSU86E) to the gravity
anomaly field in the Australian region. Based upon earlier tests, we suggest how this mean fit may be used to
estimate the ability of the model to recover AN in a particular region. Finally we recommend which model is
most suitable for use as a reference in the Australian region.
2. Testing the Geopotential Models.
2.1 Description of Technique for Testing
In earlier tests we have compared the AN derived from OSU81 against control AN, derived from (1), where AH was
found by conventional spirit levelling to 3rd-order or better, and Ah from GPS surveys (Kearsley, 1988, p, 6561).
The comparisons showed that the ability of OSU81 to recover AN varied with both location, and with the upper
limit of summation. The tests also showed that the best agreement does not necessarily occur when the
geopotential model is taken to its maximum degree and order.
To test a model for its "fitness" - its ability to recover AN, in areas where observed AN are either non-existent or
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sparse, we use the statistics of residual gravity 5g, where
5g = Ag- Ag L (2)
where Ag is the free air gravity anomaly from gravimetric survey and Ag L is the gravity anomaly generated from
the geopotential model.For these tests AgL was generated on 0.1 ° mesh across the Australian region. A value of
AgL was estimated by interpolation at each gravity point in the Australian Gravity Data Base, and 5g obtained by
(2). The data set was then analysed in 2° x 2° blocks, this approximating the area used in a spherical cap of
integration in a full gravimetric evaluation. The qSgi were then analysed to obtain the mean (m_g) and root mean
square (rms6g) for the population in the block.
This analysis was repeated for each of three recent geopotential models - OSU81, GPM2 and OSU86E. Each
refers to GRS80 (Moritz, 1980) and their maximum degree and order are 180, 200 and 360 respectively.
2.2 Inferences to be drawn from statistics
From the few tests in Australia which compared ANGrav against control AN a trend has appeared which relates
m6g to mSN. This trend is shown in Table I, where m6N is the mean fit of AN for GPS lines in the region, in
ppm of the line length, and msg is the mean (bias) of the _Sgfield in mGal, analysed over the 2° block containing
the control data.
On the basis of this evidence we have inferred across Australia the likely value of mSN from the mSg. This is of
particular importance to precise geoid studies because, as we noted above, the geopotential model will constrain
the potential precision of the full gravimetric solution if it is unable to sense the geoidal undulations at, say, the 4
to 6 ppm level or better.
3. Discussion of Results
The mean and rms of the <3gpopulation for the 256 2° x 2°blocks across the Australian continent were calculated
for each of the three models tested.
3.1 Mean fit of Geopotential models to gravity data
The values of mSg have been placed into 4 bins, as shown in Table 1. As is seen from this table, the bin limits
were chosen because they appeared to be equivalent to the 5 ppm divisions in m_5N.
The results have been summarised in the histograms in Figure I, allowing a direct comparison between the three
models.
Table 1 : Comparison of m6g with mdN
Bin + m5g 5- m6N
(reGal) (ppm)
1 0-5 0-5
2 5-13 5-10
3 13-21 10-15
4 >21 >15
(a) OSUSI and GPM2
For OSUSI, 63% of blocks lay in bin 1, 30% in bin 2, 6% in bin 3, with 2% being worse than + 21 reGal in
their mean fit. From this we infer this model can recover AN, on average, to 10 ppm or better in 93%, or nearly
24
240, of the 2 _' blocks. 'i he :uc:s _t [_)_c:< lit al_: _, :_ the: south western coax[, and all {ia,,_ccillrc ol Australia. In
the first location, the modcl may rot!cot the fact thai for the ocean regions, the Ag fiehl used in the model was
collocated from geoid undulations derived from radar altimetry. This fact may also explain the high correlation
between bin 2 and 3 areas with the coastline. The bin 3 and 4 results in Central Australia are located of the Officer
and Amadeus Basins and the McDonnell Ranges (about 1500 m elevation), an area noted for the unusually high
signal variations in the Bouger anomaly field. These results suggest the 180 degree model insufficiently sensitive
to the gravity signal in this region.
The results for GPM2 were slightly improved over those for OSUSI, reflecting the higher order of the former
model.
Figure 1: Distribution of Mean Fit of Potential Model to Terrestrial Gravity
on 2 ° x 2 ° bh)cks
Number of 2 '_ blocks = 256
5O
25
%
34%
-21 -13 -5 0
mGal
OSU81
nmax = 180
29%
10%
2%
5 13 21
50
25
39%
-21 -13 -5 0
mGal
31%
GPM2
n rmx =200
10%
1%
13 21
%
50
25
47%
32%
OSU86E
n max = 360
13%
7%
1%
-21 -_3 -5 0 5 13 21
mGal
(b) osug6E
The results of this analysis are impressive. From Figure 1 we see that 79% of the blocks fell into bin 1, 20%
into bin 2, with 1% in bin 3. No blocks had a fit worse than +_ 21 mGal. It appears that, for nearly 80% of
Australia OSU86E is capable of recovering AN to, on average, 5 ppm. In only three cases will the fit to AN be
worse than 10 ppm. It is strange that, in one of these cases OSU81 recovers Ag better than does OSU86E. There
still appears to be some correlation between bin 2 results and coastal regions, but the Fx)r bin 4 results of OSU81
on the south western coast, and in Central Australia, have disappeared. These trouble spots now lie in bin 2.
It is obvious from this analysis that the mean fit, or bias, of this model with respect to terrestrial gravity has
improved greatly over both OSU81 and GPM2. This may be explainexl by the increased order of the model,
enabling it to capture shorter wavelength features in the gravity field. The numerical analysis adopted in this
solution, which used quadratures with a desmoothing procedure suggested by Colombo (1981) may also be partly
responsible, although the fact that OSU86E summed to 180 replicates almost exactly OSU81 taken to the same
order in the South Australian and West Australian test areas tends to discount this factor.
3.2 Root mean square of residual gravity anomalies
The other statistic of significance in this analysis is the rms, which gives some measure of the fluctuations of the
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6g field from the geopotential model. The results of the computations are summarised in Table 2.
Table 2 Distribution of rms (Population 256)
Rms OSU81 GPM2 OSU86E
Bin Range (reGal)
1 0-10 64 (25%) 62 (24%) 143 (56%)
2 10-20 142 (55%) 151 (59%) 96 (37%)
3 20-30 38 (15%) 32 (13%) 15 (6%)
4 >30 12(5%) 11 (4%) 2(1%)
As is seen from these figures and the summary, OSU81 and GPM2 perform nearly equally well, with 80% and
83% of hlocks with rms of less than 10 reGal. As with the mean fit, the blocks of poorest representation are
generally m the south western corner and in the centre, and along the east coast of Australia. Table 2 also shows
how much the 36)0 degree model recovers the variations in the gravity field more faithfully. Over half the area is
recovered with an rms of less than 10 reGal, while all but 7% of the blocks have an rms less than 20 reGal. Only
2 blocks modeled by OSU86E have an rms > 30 mGal.
Combining mean and rms for OSU86E
The above results confirm the integrity of OSU86E, already demonstrated in the test on the mean fit, in its ability
to recover the gravity field across Australia. Indeed, when combining the mean and the rms results for OSU86E,
we find that a large portion of the region have both small bias and rms values. Over 40% of the blocks analysed
have a mean and an rms both of which lie in bin I. Obviously the shorter wavelength features in these areas are
well modeled by OSU86E. From Table 1 we infer that, in these regions, OSU86E will recover AN to better than
5 ppm, and that there may be little benefit in incorporating the short wavelength signal in AN obtained from the
detailed analysis of surface gravity. Predictably, however, most of these areas lie across the inland, sparsely
developed region of Australia. The areas which contain most development activity, the coastal regions, still
exhibit less favourable recovery by this model and will apparently still require a full gravimetric solution for
highest precision.
4. Conclusions
From these tests we see how the ability of a geopotential model to fit the gravity field across Australia improves
with the increased order of the model Iit appears that this improvement is due almost entirely to the higher order of
the model, and not to the different numerical technique used to solve for the potential coefficients. Finally, it
appears that for 40% of the region, OSU86Ewill serve in the recovery of AN for all but the most exacting
purposes. However for the coastal regions where most development activity occurs, a full gravimetric solution
involving a detailed analysis of detailed gravity will be required for most higher-order surveying tasks.
References
Colombo, O.L., Numerical methods for harmonic analysis on the sphere, Report No. 310, Dept. Geod. Sci., The
Ohio State University, Colombus, 1981.
Kearsley, A. H. W., Tests on the recovery of precise geoid height differences from gravimetry, J. Geophys. Res.,
93(B6), 6559-6570, 1988
Moritz, H., Geodetic Reference System 1980, Bull. Geod., 54.(3), 395-405, 1980.
Rapp, R. H., The earth's gravity field to degree and order 180 using Seasat altimeter data, terrestrial gravity data
and other data, Rep. 322, Dep. of Geod. Sci. and Surv., Ohio State.
Rapp, Richard H. and Cruz, Jaime, Y. Spherical harmonic expansions of the earth's gravitational potential to
degree 360 using 30' mean anomalies, Report No. 376, Dept Geod. Sci. and Surv., The Ohio State University,
Columbus, Ohio, 1986.
Wcnzcl, H.-G. Hochaufl6sendc Kugelfunktionsmodelle ftir dos Gravitationspotential dcr Erde, Wissenschaftliche
Arbeiten der Fachrichtung Vermessungswessen der Universitdt llannover, Hannover, 1985.
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Geopotential models in the Australian region

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