&2~6 

C& Reports of the Department of Geodetic Science and Surveying 

Report No. 376 

Spherical Harmonic Expansions of the Earth's Gravitational Potential To Degree 

360 Using 30' Mean Anomalies 

by 

Richard H. Rapp 

and 

Jaime Y. Cruz 

(LASA-CB-18025 1) SPB ERIC AL EAF BC BIC 887-18937 

EXPANSICBS 03 2fiI EAR3E'S GEB\1!IAIXOHAZ 

FCTEP'IIAL TO CEGREE 360 USIbG 3C' BEAN 

BYCf!lALXES (Ohio State Uriiv,) 25 & CSCL 0% Unclas 

G3/46 43806 

Prepared for 

National Aeronautics and Space Ad ministration 

Goddard Space Flight Center 

Greenbelt, Maryland 20770 

NASA Grant No. NGR 36-008-161 

OSURF Project 783210 

, The Ohio State University 

Department of Geodetic Science and Surveying 

1958 Neil Avenue 

Columbus, Ohio 43210-1247 

December 1986 
Abstract 

Two potential coefficient fields that are complete to degree and order 360 

have been computed. One field (OSU86E) excludes geophysically predicted 

anomalies while the other (OSU86F) includes such anomalies. These fields were 

computed using a set of 30’ mean gravity anomalies derived from satellite 

altimetry in the ocean areas and from land measurements in North America, 

Europe, Australia, Japan and a few other areas. Where no 30’ data existed, 

l*xl* mean anomaly estimates were used if available. No rigorous combination 

of satellite and terrestrial data was carried out. Instead we took advantage of 

the adjusted anomalies and potential coefficients from a rigerous combination 

of the GEMLZ’ potential coefficient set and l*xl* mean gravity anomalies. 

The two new fields were computed using a quadrature procedure with 

de-smoothing factors suggested by Colombo. The spectra of the new fields 

agree well with the spectra of the fields with l*xl* data out to degree 180. 

Above degree 180 the new fields have more power. The fields have been 

tested through comparison of Doppler station geoid undulations with 

undulations from various geopotential models. The agreement between the two 

types of undulations is approximately *1.6 m. The use of a 360 field over a 

180 field does not significantly improve the compariosn. Instead it allows the 

comparison to be done at some stations where high frequency effects are 

important. In addition maps made in areas of high frequency information 

(such as trench areas) clearly reveal the signal in the new fields from degree 

181 to 360. 

ii 
1 

Foreword 

This report was prepared by Richard H. Rapp, Professor, and Jaime Y. 

Cruz, Post-Doctoral Researcher, Department of Geodetic Science and Surveying, 

The Ohio State University, under NASA Grant NGR36-008-161, The Ohio State 

University Research Foundation Project No. 783210. The grant covering this 

research is administered through the NASA Goddard Space Flight Center, 

Greenbelt Maryland 20771. The NASA Technical Officer for this grant is Mr. 

Jean Welker, Code 621, 

Some computer funds for the research described here were supplied 

through the Instruction and Research Computer Center. 

.. The reproduction and distribution of this report was carried out with 

funds supplied, in part, by the Department' of Geodetic Science and Surveying. 

iii 
1. Introduction 

This report describes our first attempt at carrying out a spherical 

harmonic expansion of the earth's gravity field to degree and order 360. The 

procedure followed is built on an adjustment of satellite derived potential 

coefficients and l'xl'. mean gravity anomalies that is described by Rapp and 

Cruz (1986). This combination solution yielded a set of potential coefficients 

corresponding to those values with a priori information, and a set of adjusted 

l-xl- anomalies. These anomalies were then used in a least squares collocation 

estimation procedure to determine a set of potential coefficient to degree 250. 

Examination of the spectrum of this field led to the suggestion that smoothing 

was taking place at the higher degrees due to the averaging in the l'xl' 

cells. 

In order to estimate more reliable high degree fields we decided to use 

anomalies in smaller block sizes--specifically 30'x30' anomalies. A rigorous 

repeat of the prior adjustment was not possible so that an alternate procedure 

was developed to obtain an expansion to degree 360. The following sections 

describe the methods used and the results obtained. 

2. The Previous Combination Solution 

In this section we briefly review the procedures that were followed by 

Rapp and Cruz (1986). The starting equation is the expression for the 

gravitational potential written in the following form: 

where: 

r,+,A geocentric coordinates 

kM geocentric gravitational constant 

a 

CQm, Sam 

Pem 

equatorial radius of the reference ellipsoid 

fully normalized potential coefficients 

fully normalized Legendre function of degree 1 and order m. 

- 

- 

The disturbing potential can be written as: 

1 
where: 

The gravity anomaly, in the Molodensky surface free-air anomaly sense is 

given as: 

where P is at the surface and Q is on the telluroid. The boundary condition 

that relates Ag and T is as follows: 

where h is the plumbline direction. Neglecting deflections of the vertical 

equation (2) can be substituted into (5) to obtain: 

-d, i where Cam (i=h,7) are ellipsoidal corrections. If we assume that we have the 

anomalies (Ag) given on the surface of the reference ellipsoid equation (6) can 

be inverted to solve for potential coefficients (Cruz, 1986). We have: 

In (7) we have: 

2 
The other terms in (7) are ellipsoidal correction terms defined in Rapp and 

Cruz (ibid, pp. 16-17). 

The gravity anomaly on the ellipsoid must be determined by downward 

continuing the surface anomaly to the ellipsoid and adding an atmospheric 

correction (SgA). We have used the following: 

This downward continuation is an approximation. For our work the derivatives 

were calculated from an expansion to degree 300 which was developed as part 

of a previous solution (Rapp, 1981). 

The combination solution was carried out through the adjustment of the 

potential coefficients derived from satellite analysis and the coefficients 

implied by gravity data. Specifically the mathematical model, F, was taken as 

F = Lxa - Lca = 0 

where L,a. is the adjusted potential coefficient set derived from satellite data 

and Lxca is the potential coefficient set implied by the adjusted gravity 

anomalies given in cells on the earth. The linearization of (10) is written: 

where VQ are corrections to the approximate anomalies, arid Vx are corrections 

to the a priori known potential coefficients. The W vector is: 

w = ho - hoc 

where &o is the approximate potential coefficient set and L, c is the 

potential coefficient set implied by "observed" anomalies. The model is 

formulated taking into account ellipsoidal corrections and downward 

continuation effects. 

given potential coefficient using (7). We have: 

0 

-d, 0 We start this by computing the Cjm values from the 

-d 0 where AC& 

equation (8) the values of hoc are computed. 

are all but the first terms on the right side of (7). Using 

3 
The adjustment is now carried out to obtain V, and VQ. Specifically: 

v, = - ((Be Pjjl Be')-' + PX)-' (Bg PI1 B1)W (14) 

PQ is the weight matrix of the observations and Px is the weight matrix of the 

a priori potential coefficients. The elements in BQ are given by the 

coefficients of AgE in equation (8). The adjusted potential coefficients, 

compatible with satellite derived estimates, would be: 

The adjusted anomalies on the ellipsoid would be: 

LQa = LQO + VQ (17) 

while the adjusted anomalies at the surface of the earth would be found from 

equation (9). 

In our combination solution we used the GEMLZ' potential coefficients that 

are very similar to the GEMLZ coefficients described by Lerch et al. (1982). 

In addition we used a few additional coefficients to degree and order 30. The 

total number of coefficients estimated was 582. 

The loxlo gravity anomaly data was based on the merger of terrestrial 

anomaly estimates with altimeter derived gravity anomalies. In this merger we 

considered two forms. One form excluded most geophysically predicted 

anomalies while the second form included such anomalies. In the first case we 

had a total of 50,562 l*xl* anomalies while in the second case there were 

56,109 values. Figures 3 and 4 of Rapp and Cruz (ibid) show the location of 

these anomalies. In order to fill in the "empty" areas we computed the 

anomalies from the a priori potential coefficient set. 

The higher degree potential coefficient fields were made by applying an 

optimal estimation procedure to the adjusted anomalies on the ellipsoid. This 

would correspond to a least squares collocation estimation using appropriate 

covariance functions, anomaly degree variance models, and anomaly accuracy 

estimates. The process of the optimal estimation as developed by Colombo 

(1981) with implementation for l*xl* data described by Hajela (1984) requires, 

for a complete solution, extensive computations. The solution we estimated was 

to degree'250. 

Acem corrections introduced in equation (13). 

After the coefficients are estimated it is necessary to add the 

-d,O 

4 
It is possible to approximate the optrmal estimate through the use of a 

modified orthogonality procedure. 

complete surface of the ellipsoid we could use (8) to compute C& . Since 

data are actually given in discrete cells an implementation of(8), to 

approximate an optimum estimation, was suggested by Colombo (1981). If the 

data is given in blocks of 8' let N = 180'/8'. 

If the anomalies were given on the 

--d 0 

Then we have: 

where: y = kM/az; 

AZi j = the mean anomaly corresponding to AgE; 

Be is the Pellinen smoothing operator that depends on a circular cap radius 

which in turn depends on the size of the mean anomaly cell (Rapp and Cruz 

(ibid, p. 22)). 

Using the adjusted l'xl- mean anomalies of our combination solution we 

computed a set of potential coefficients from equation (18) which we have 

compared to the values from the optimal estimation solution. Table 1 (taken 

from Rapp and Cruz (ibid, Table 11) shows the differences in terms of 

undulations, anomalies, and percentage d3ferelices. 

Table 1. Comparison of the Potential Coefficient Solution from Equation (18) 

with the Optimal Estimate Using l-xl- Mean Anomalies. 

B 6N (a) Ibg(mga1s)l P(%) 

2 0.0 0.00 0.0 

10 0.0 0.00 0.0 

20 0.0 0.00 0.0 

30 0.2 0.01 0.5 

50 0.2 0.02 1.1 

75 0.8 0.09 6.2 

100 1.0 0.15 10.6 

120 1.0 0.18 13.2 

150 0.6 0.14 13.0 

180 0.8 0'. 22 22.3 

to 180 8.9 1.53 7.6 

5 
We see from this table that there is good agreement between the two solutions. 

The percentage differences are negligible at the lower degrees increasing to 

22% at degree 180. The results from this test will be used later to justify the 

use of equation (18) with 30’ mean anomalies instead of using the complicated 

optimal estimation results. 

This section has been written to describe our potential coefficient 

solution using l’xl’ anomaly data and the GEMLB’ potential coefficients. The 

results of these computations were two potential coefficient sets computed 

using an optimal estimation procedure with a uniform data noise of one mgal 

for the anomalies. The solution that excludes geophysically predicted 

anomalies was designated OSU86C; the solution that included such anomalies 

was called OSU86D. Both solutions were complete to degree and order 250. 

3. The 30’x30’ Mean Anomaly Data Set 

In order to extend the OSU36C/D solutions to a higher degree we first 

needed to put together a set of smaller anomaly cells in order to get the 

higher frequencies into our solution. The cell size chosen was 30’x30’ 

although other sizes could be chosen. However this size is convenient to 

work with as four 30’x30’ cells fill a l*xlo cell. 

The 30’x30’ mean anomalies for the ocean areas had been computed by 

Rapp (1986) from Geos-3/Seasat altimeter data. Specifically computed were 

point anomalies on a 1/8’ grid which were meaned to compute both 30’ and 1’ 

mean anomalies (and sea surface heights). The total number of predicted 30’ 

anomalies was 149, 670 although about 1000 were considered unreliable because 

they were on land or in areas in which the altimeter data was sparse. 

In order to obtain data from terrestrial ’estimates it was necessary to 

collect gravity anomaly data as point values or small (lO’xlO’, 6’xlO’) mean 

values. Such values were then formed into 30’x30’ values with individuai 

accuracy estimates where possible. The development of this 30’ terrestrial 

anomaly data base is described by Despotakis (1936). Figure 1 shows the 

location of the 30’ mean anomalies in the August 1986 field. 

We next created a merger of the altimeter derived anomalies and the 

terrestrial values. In the ocean areas we used the altimeter derived anomaly 

unless its standard deviation was greater than 30 mgals or if the anomaly was 

in a l’xl. block whose elevation was greater than or equal to zero. For the 

remaining areas the 30’ terrestrial values were used if they were available. 

The atmospheric correction (6gA) was added to the terrestrial anomalies (see 

equation 9) and a gravity formula correction of -0.6 mgal was added to 

convert to our reference system of constants. If neither an altimeter derived 

anomaly nor terrestrial anomaly was available the adjusted l’xl’ anomaly from 

the combination solution was used as fill-in values. 

6 
0 

m - 

0 

2 

a 

=c - 

c 

N - 

0 

0 - 

0 

03 

0 

(D 

0 

7 

0 

N 

0 

0 

s 

0 

0 

N 

0 

0 

0 

m 

0 

N 

m 

0 

a 

N 

0 

s 

N 

0 

N 

N 

0 

O 

N 

3 

2 

c 

.C( 

m 

rn 

m 

Q) 

c .- 

8 

E 

4 

4 

& 

.C( 

0 

c3 

. F4 

Q) 

L 

3 

ka 

E 

.d 

7 
a 

z 

2 

z 

2 

a, 

0 

0 

0 

0 

0 

OJ 

0 

(D 

0 

? 

0 

N 

0 

0 

=r 

0 

0 

N 

m 

0 

0 

m 

0 

0 

(u 

0 

(D 

cu 

0 

3 

cu 

0 

cu 

cu 

0 

0 

(u 

/ 

0 

a - 

8 
In order to take advantage of the least squares adjustment that led to the 

OSU86C/D solutions we took the adjusted l’xl’ values of these solutions and 

forced the mean of the four 30’ values within the blocks to be the same by 

applying a bias term to each of the non-fill-in 30’ values. Specifically we 

write: 

In (20) the A stands for adjusted and UA stands for unadjusted and n is the 

number (usually 4) of non-fill-in 30’ values in the 1’ blocks. The B term 

would consist of three effects: the downward continuation of the surface 

anomalies to the ellipsoid (see equation 9); the inconsistencies of the 30’ and 

1’ values; and differences caused by the adjustment process. The B value for 

each 1’ block was then added to the non-fill-in 30’ value to obtain a value 

that we consider to be on the ellipsoid (AgE) and is consistent with the 

adjusted anomalies of our earlier solutions. 

In summary the merger process used 139,946 values derived from altimeter 

data; 21,739 values from the terrestrial data, and 97,515 “fill-in” values from 

the adjusted l-xl- data. The total number of 30’ anomalies is 259,200. Figure 

2 shows the location of 40700 1-xl’ blocks in which at least one 30’ value from 

either the altimeter derived or terrestrial data set was used. 

4. The Estimation of the Potential Coefficients 

If we were to start the combination process from our 30’ data we would 

need to evaluate equations (14) and (15). This process is feasible with a 

vector processing machine. iiowever we did noi feel that the effort far such 

a solution was worthwhile at this time. Since the actual adjustment is done 

with the coefficients of the satellite solution, which was complete only to 

degree and order 20, we felt that whether this adjustment was made with 1’ 

data or 30’ data would be irrelevant. We consequently adopted the adjusted 

coefficients of the OSU86C/D solutions. For the other coefficients to degree 

360 we used equation (18) through the HARMIN program of Colombo (1981) 

where the AZi values were taken as the bias corrected 30’ values. This 

computation required an evaluation of the integrals of the fully normalized 

associated Legendre functions up to degree and order 360 at 30’ intervals. 

This was done using program F428AV1 which implements the Paul (1978) 

subroutine with changes suggested by Gleason (1985). 

The HARMIN program (OSU program F419B) was used with the qa factors 

given by equation (19). Two solutions to degree 360 were carried out. The 

OSU86E solution basically excluded geophysically predicted anomalies while the 

OSU86F solution included such anomalies. 

from equation (18) were found, the ellipsoidal corrections given by Acem of 

equation (13) were applied. These corrected coefficients were then merged 

with the adjusted and corrected coefficients of the OSU86C or D solution to 

form the final potential coefficient sets. 

After the potential coefficients 

-&, 0 

9 
5. Potential Coefficient Accuracy 

One disadvantage of using the HARMIN approach to the potential 

coefficient estimation is the lack of estimated coefficient accuracies. In fact, 

the use of the more rigorous optimal estimation procedure does not ensure 

realistic accuracies because of the smoothing of the field that takes place at 

the higher degrees when realistic anomaly standard deviations are used. To 

avoid this problem Rapp and Cruz (ibid) combined the sampling error (from , 

the optimal estimation procedure) with a propagated noise assuming 

uncorrelated anomaly errors of 10 rngal. This procedure led to a noise 

magnitude that was equal to the signal near degree 175 (see Figure 3). Since 

we do not have a formal sampling error estimate for the 30' solution, and 

since the assumption of uncorrelated 30' anomaly errors is even more 

unrealistic than the 1' case we have not computed the potential coefficient 

errors for the OSU86E and' F solutions. If there are requirements that 

demand coefficient accuracy estimates for the 86E/F fields, we suggest the use 

of the 86C/D standard derivation to degree 175 followed by a 100% uncertainty 

in the coefficients above that degree. 

6. Anomaly Degree Variances 

We first define the spectrum or power at degree 1 as follows: 

The anomaly degree variances, CQ are given as: 

where 7 = kM/a'. The anomaly degree variances are formally given on a 

sphere of radius a. Figure 3 shows the anomaly degree variances for the 

OSU86C and the OSU86E solution. Also shown is the accuracy estimate of the 

OSU86C solution based on the optimal estimation procedure with an anomaly 

standard deviation of one mgal and a propagated standard deviation of 10 

mgal for the l'xl* anomalies (Rapp and Cruz, ibid, equation (6.11)). We see 

that the power is almost identical in the two solutions out to degree 200 

although OSU86C solution shows slightly less power between degrees 120 and 

160. Beyond degree 204, out to degree 250, the 86C solution has power that 

falls off much more rapidly than the solution (OSU86E) with the 30' anomalies. 

This seems to reflect the loss of information, not recovered by the optimal 

estimation procedure, in the 1' averaging process. This might imply that the 

rough rule of thumb (180'/0', 9' = block size) for how high a degree should a 

spherical harmonic expansion be taken may not be unreasonable. 

10 
0 

0 

0 
0 0 oov, cu rlv) N dm Nrc v, cui--( 

00 0 00 

'0 00 

u) cu A. *o 

0 

0 

0 

.O 

=;. 

0 

0 

0 

.m 

m 

. 

0 

0 

0 

0 

m 

. 
Figure 3 also shows the anomaly degree variance implied by the hula rule 

for the decay of potential coefficients (i.e. the 10-s/12 rule). The anomaly 

degree variance implied by this rule has been computed on a sphere of radius 

a assuming the original rule was based on data referred to the mean surface 

of the earth. Specifically plotted was: 

cl(a) = - l+1.5 

where RE = 6371 km and a = 6378 km. The rule has too much power between 

degrees 10 to 50 and too little power between degrees 80 to 200. Beyond 200 

the estimates from the model and from OSU86E are within 50% of each.other. 

Figure 4 shows the anomaly degree variances for the OSU86D (1') and the 

OSU86F (30') solutions. Both these solutions used the geophysically predicted 

anomalies. Also shown is the accuracy of the OSU86C/D solutions. The 

comments made with respect to the anomaly degree variances shown in Figure 

3 are also applicable here. 

Specific values of the anomaly degree variances, at selected degrees, for 

the four solutions discussed here are given in Table 2. 

Table 2. Anomaly Degree Variances. (Units are mgall). 

Degree 

1 

50 

100 

150 

180 

200 

250 

300 

350 

From this table we see that there is no substantial difference, at the higher 

degrees, in the power of the OSU86E and F solutions. 

7. Doppler Undulation Comparisons 

Solution 

OSU86C OSU86E OSU86D OSU86F 

2.66 2-5? 2.94 2.83 

2.23 2.13 2.35 2.23 

1.36 1.28 1.48 1.33 

0.96 0.97 1.02 1.01 

0.57 0.60 0.56 0.61 

0.14 0.41 0.16 0.43 

0.22 0.22 

0.18 0.18 

. As discussed by Rapp and Cruz (ibid, section 8.3) the accuracy of a 

.potential coefficient model may be judged by comparing the geoid undulation 

derived from the model and the value implied by the ellipsoidal and 

orthometric heights at a station. This can only be done after the station 

coordinates have been converted to a geocentric, true scale system, and the 

13 
best estimate of the equatiorial radius is used. 

Mean Std Dev 

Model 4(max) Difference Difference 

OSU8 1 180 -1.18 m 11.57 m 

GPM2 180 -0.96 1.51 

OSU86C 180 -1.13 1.60 

osU86D 180 -1.12 1.60 

OSU86E 180 -1.13 1.58 

OSU86E 360 -1.10 1.57 

OSU86F 180 -1.12 1.58 

OSU86F 360 -1.09 1.57 

We carried out these computations with the new models for a global 

Doppler station set, and data sets in North America, Australia and in Europe. 

In these tests an equatorial radius of 6378136 m was used. the mean 

difference (Doppler minus model) and the standard deviation of the difference 

is given in Table 3 for Europe, and Table 4 for Australia for a fixed number 

of stations. 

I 

Table 3. Comparison of Doppler Derived Undulations in Europe With Values 

From Models. 173 Stations Used. 

Mean Std Dev 

Model 4(max) Difference Difference 

OSU81 180 -.04 m e1.51 m 

GPM2 180 -. 28 1.47 

OSU86C 180 -. 47 1.38 

OSU86D 180 -. 39 1.42 

OSU86E 180 -.48 1.35 

OSU86E 360 -.50 1.33 

OSU86F 180 -.40 1.38 

OSU86F 360 -.42 1.36 

Table 4. Comparison of Doppler Derived Undulations in Australia With Values 

From Models. 114 Stations Used. 

From these two tables we see that in Europe, of the degree 180 solutions the 

OSU86E is best, while in Australia GPM2 is the best. The degree 360 fields in 

Europe show a slight improvement over their 180 counterparts. In Australia 

the 360 fields are slightly better than the 180 fields with the exception of 

GPM2 which fits the best for this area. 

For North America and the global data set we carried out the comparisons 

rejecting stations where the residual was greater than 4 m. Therefore in 

14 
judging the results of these comparisons we consider the standard deviation 

- and the number of stations accepted for the comparison. These results are 

shown in Table 5, for North America, and in Table 6, globally. 

Table 5. Comparison of Doppler Derived Undulations in North 

Values from Models 

Mean Std Dev Num of 

Model e(max) Difference Difference Stations 

OSU81 180 0.22 11.72 m 683 

GPMZ 180 0.18 1.58 695 

OSU86C 180 0.20 1.54 687 

OSU86D 180 0.26 1.55 687 

OSU86E 180 0.21 1.52 689 

OSU86E 360 0.26 1.45 695 

OSU86F 180 0.28 1.52 688 

OSU86F 360 0.31 1.47 696 

America With 

Table 6. Comparison of Doppler Derived Undulations Globally With Values From 

Models. 

Mean Std Dev Num of 

Model 1 (max) Difference Difference Stations 

OSU81 180 0.18 m *1.76 m 1721 

G?M2 180 0.13 1.64 1735 

OSU86C 180 0.17 1.66 1741 

OSU86D 180 0.21 i.66 i743 

OSU86E 180 0.16 1.67 1752 

OSU86E 360 0.17 1.66 1777 

OSU86F 180 0.21 1.66 1754 

OSU86F 360 0.21 1.66 1780 

From Table 5 we see, of the 180 solutions, both the OSU86E/F solutions 

give essentially the same results. Improved results are found with the 360 

solutions with a small decrease in the standard deviation of fit and a small 

increase in the number of stations accepted. 

From Table 6 we see all the solutions except the OSU81 solution give about 

the same standard deviation of fit. Of the 180 solutions, the OSU86F solution 

accepts the most stations. Of the two 360 solutions tested the OSU86F solution 

accepts the most stations. 

In judging these results we must recall that the accuracy of the Doppler 

derived undulations may not be sufficient to distinguish the accuracy of 

various models at or below the t1.5 m level. More accurate results might be 

obtained using laser station coordinates. Unfortunately such stations are 

15 
much fewer than the Doppler stations and they are located at difficult 

locations (e.g. mountain tops). 

Rapp and Cruz (ibid, Section 8.4) discussed the undulation residual 

correlation with topography. Such correlation continues for these 360 fields. 

For example for the OSU86F field to degree 360, with the global station set, 

the slope is (0.411.14) m/km with 1780 stations accepted. The value for the 

OSU86D field to degree 180 is (0.412.12) m/km with 1741 stations accepted. 

8. High Degree Geoid Maps 

It would be possible, but not really meaningful, to compute a global geoid 

undulation map using the two new potential coefficient models to degree 360, . Such maps would not be meaningful because the high degree contributions to 

the undulation, for the most part, ake quite small. To first estimate the 

magnitude of the high degree terms we computed, for selected degrees, the 

undulation contribution from degrees i+1 to degree imm. Values for selected 

degrees are given in Table 7. 

Table 7. Geoid Undulation Magnitudes for Selected Degree Ranges in the 

OSU86F Model 

Undulation Contribution 

180 to 250 120.2 Qn 

181 to 360 23.2 cm 

251 to 300 

The values given in Table 7 represent global averages. To see specific 

estimates of the contribution made by the higher degree terms, we prepared 

two maps showing the geoid undulations computed from the coefficients from 

degree 181 to 360. Figure 5 shows this map for a region in the United States 

covering a portion of the Rocky Mountains. Figure 6 shows these effects for 

a region over the Tonga Trench. 

In the Rockies test we can see effects that reach 1.8 m. In the Tonga 

Trench area the high frequency signal over the trench is clearly visible in 

the lineated pattern. The largest values are on the order of 2 meters. The 

latter case shows how the high degree fields can play a role where high 

frequency information is present in the gravity field. Note from Figure 6 

that, outside the trench the contributions of the higher frequencies are quite 

small, approaching, it is speculated, the *23 cm computed for Table 7. 

Forsberg (1986) has computed the geoid undulation information above 

degree 180 based on power spectral analysis of gravity anomalies in Norway, 

Sweden, Finland and Denmark. Averaging over 37 2.~4' areas the information 

above degree 180 is *32 cm with uncorrected anomaly data and *28 cm with 

terrain corrected anomaly data. These values compare well with the 123 cm 

value given in Table 7 when the summation is taken just to degree 360. 

16 
Figure 5. Geoid Undulations in a Rccky Mountain Region from Degrees 181 to 

360 of the OSU86E Solution. The contour interval is 0.5 m. The 

data grid was 0.'25. 

Y3' 

r2' 

Y 1' 

Y 0' 

39' 

38' 

37' 

36' 

3s' 

45' 

Y '1O 

Y3O 

Y 2' 

Y I' 

rlo' 

39' 

38' 

37' 

36' 

3s' 
Figure 6. Geoid Undulations in the Tongs Trench Area from Degrees 181 to 

360 of the OSV86E Solution. The contour interval is 0.5 m. The 

data grid was 0.'25. 

- 1s' 

-16" 

- 17' 

-16' 

-19' 

-20" 

-21" 

- 22" 

- 23' 

-2r' 

-2s' 

e 0 e 0 0 0 e 0 

m u3 h 43 Q) 0 

CD 43 43 43 43 Q) 

4 4 4 4 1 4 

- 0 e 0 

G 0 -4 eJ m 

(D 43 a (D 

4 4 4 4 4 

1 1 4 

0 

0 

Q, 

rl 

- 15' 

- 16' 

- 17' 

- 18" 

- 19' 

-zoo 

-21' 

- 22' 

- 23' 

-2y0 

-25' 

18 
9. Coefficient CornDarisons 

It is of interest to compare the potential coefficients of the C/D solution 

with coefficients of the E/F solution up to degree 180. The C vs E comparison 

is given in Table 8 while the D/F comparison is given in Table 9. 

Table 8. A Comparison of the OSU86C and E Models at Selected Degrees 

1 6X( m) Gg(mga1s 1 Percent Diff. 

50 0.7 0.05 3.3 

75 0.9 0.11 6.9 

100 1.2 0.18 11.9 

120 1.4 0.26 17.2 

150 1.7 0.38 33.0 

180 1.8 0.49 50.2 

2 to 180 15.7 3.18 14.6 

Table 9. A Comparison of the OSU86D and F Models at Selected Degrees 

a 6N3(cx1) Gg(mgals1 Percent Diff. 

50 0.8 0.06 3.5 

75 1.0 0.11 6.9 

100 1.2 0.18 11.7 

120 1.5 0.27 17.9 

150 1.7 0.40 32.8 

180 1.8 0.49 48.9 

2 to 180 16.1 3.25 14.6 

The main thing that we see from these two tables is that the differences 

between the 1' and 30' data solutions are small up to degree 180. The 

differences are a function of degree, with the largest differences occuring at 

the higher degrees. This gives us some confidence in the 1' solutions but it 

also shows us the coefficients at the higher degrees may change by 50% from 

a 1' solution to a 30' solution. It would not be unreasonable to expect the 

coefficients at the high degrees of the E/F fields to change by this amount if 

0.'25 anomalies were used instead of 0.'5 values. 

As a last comparison we show in Figure 6 the undulation differences 

between the OSU86F and the OSU81 solutions up to degree 180. The contour 

map has been created from data given on a 2.~2' grid so that some high 

frequency differences may be missing. The maximum difference is 9.7 m which 

occurs in the southwest part of Africa. Most of the large .undulation changes 

are due to substantial anomaly differences in the solutions. 

19 
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10. Conclusions 

This report has described the estimation of an expansion of the earth's 

gravitational potential to degree 360. The method used, did not rigorously 

combine, satellite and terrestrial gravity data. Instead, we built on solutions 

that were formed using l'xl' anomalies. We used a 30' data base formed from 

the merger of the anomalies derived from satellite altimetry in the oceans and 

terrestrial data for some land areas. Many land areas have no coverage or 

coverage only in terms of l'xl- anomaly estimates. In such areas the field 

will not represent the high frequency variations that may actually exist in the 

area. 

An important part of the anomaly reduction process is the downward 

continuation of the surface anomalies to the ellipsoid. In dealing with the 1' 

data we used a two term Taylor series involving a preliminary high degree 

field. We did not compute the downward continuation effects for the 30' cells. 

Instead we effectively used for each 30' cell the value used in the 1' 

reductions. Future solutions should examine improved techniques to carry out 

this downward continuation process. 

We did not compute accuracy estimates for the coefficients. Such 

estimates would require a number of assumptions that would be unrealistic in 

practice. More work needs to be done in this area. 

The next generation of the 360 field could involve the formal adjustment 

of the 30' mean anomalies and satellite data. More complete downward 

continuation procedures could be attempted. Accuracy evaluation should be of 

high importance. 

Although there are a number of improvements that could be made for the 

next solution, the current solution appears to represent the given data and 

exter~al data (e.g. Doppler unbult?tinr?s) =Trite well. We must. remember, that 

just because we have a high degree field, it does not mean we have a highly 

accurate high degree field. The latter type of field will only be available 

when more precise data, such as would come from the Geopotential Research 

Mission of NASA (1986) would become available. Until then our solutions to 360 

can be used for a number of purposes including realistic simulations for the 

GRM mission. . 

21 
References 

Colombo, O.L., Numerical Methods for Harmonic Analysis on the Sphere, Report 

No. 310, Dept. of Geodetic Science, The Ohio State University, Columbus, 

1981. 

Cruz, J.Y., Ellipsoidal Corrections to Potential Coefficients Obtained from 

Gravity Anomaly Data on the Ellipsoid, Report No. 371, Dept. of Geodetic 

Science and Surveying, The Ohio State University, Columbus, August 1986. 

Despotakis, V., The Development of the June 1986 l*xl* and the August 1986 

30’x30’ Terrestrial Mean Free-Air Anomaly Data Bases, Internal Report, 

Dept. of Geodetic Science and Surveying, The Ohio State University, 

Columbus, OH, 1986. 

Forsberg, R., Spectral Properties of the Gravity Field in the Nordic Countries, 

1986. 

Gleason, D., Partial sums of Legendre series via Clenshaw summation, 

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Lerch, F.J., S. Klosko, and G. Patel, A Refined Gravity Model from Lageos 

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Paul, M.K., Recurrence Relations for Integrals of the Associated Legendre 

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22