Evaluation of the EGM96 Model of the Geopotential in the United States
(Addendum to:
Evaluation of Preliminary Models of the Geopotential in the United States
)
D. A. Smith and D. G. Milbert
National Geodetic Survey, NOAA
Introduction
As a final report for the
Special Working Group
of the International Geoid Service, we
have evaluated the geopotential model EGM96. The tests are all similar to those performed on
models EGM-X01 through EGM-X05, and comparisons between the final model and the
preliminary models are made. The EGM96 model served as the base model in the remove-compute-restore
procedure in the calculation of
G96SSS
, the recently released high-resolution
model of the geoid for the United States (and the foundation of the
GEOID96
product).
It must be noted that some of the data sets used in this evaluation are not the final data
sets used in the computation of G96SSS. These older data sets were used in this evaluation
strictly to maintain compatibility with the previous beta test report (Smith and Milbert 1997).
Use of EGM96 Coefficients in Computing Geoid Heights
Between the time of the beta report and the testing of EGM96, some clarifications were
made regarding the computation of geoid undulations from a spherical harmonic model of the
Earth's external gravitational potential. Specifically, the treatment of the degree zero component
of the geoid undulations was investigated. It was pointed out that a correction for the difference
between the normal potential of GRS80 and the true potential of the geoid (as best we know it),
should be applied to geoid undulation computations, when using GRS80 as the chosen reference
field. However, in an effort to maintain compatibility with results in the beta model report, we
will continue to neglect this correction in this report. However, it has been implemented in the
computation of GEOID96. The effect of this term amounts to a constant bias of 41 cm in the
geoid undulation computations.
Low Resolution Geoid Computation and Evaluation
Using the methods described in the previous section, combined with the height anomaly
to geoid undulation correction (the "up/down" procedure) discussed in the beta report, geoid
undulations were computed from EGM96, incorporating a 3'x3' ellipsoid height
Digital Terrain Elevation Data (DTED) for the
conterminous United States, and a 3'x3' grid of simple Bouguer anomalies. This grid of geoid
undulations (NEGM96) was compared to the geoid undulations implied by 2497 GPS ellipsoidal
heights on leveled benchmarks (NBM). A tilted plane was fit through the residuals
(NEGM96 - NBM)
in the same way as the beta models, and the results of that fit are shown below.
Model Offset(cm.) Tilt(ppm) RMS about plane(cm) Azimuth(deg) X01 -2.16 0.40 26.52 338 X02 1.02 0.32 29.77 336 X03 0.07 0.35 26.22 334 X04 0.43 0.35 25.99 335 X05 0.76 0.35 26.10 335 EGM96 2.01 0.41 27.03 343
These statistics show EGM96 is something of an outlier, relative to the five beta models,
without being unreasonably different. In order to help illustrate the continental scale of the tilt,
the following figures have been provided.
Figure 1
shows a scatter plot of geoid undulation
residuals relative to orthometric height. It is encouraging to see that there is little dependence
upon elevation, though a small correlation could possibly be inferred.
Figure 2
shows the scatter plot of geoid undulation residuals relative to latitude, and
Figure 3
shows them relative to
longitude. The combined message of these two plots is that a northerly and westerly (i.e.
azimuth of 343) upward tilt is clearly occurring across the entire United States, and is not just
being driven by any cluster of localized residuals. Please keep in mind that these statistics do not
reflect the final data sets and theory used in G96SSS, but rather are being kept internally
consistent with the report on the beta models.
Although it is important to maintain consistency with Smith and Milbert (1997) (for
the rest of this paper, called the "beta report"), it is also quite useful to see how EGM96
compares to our final, best gravimetric geoid model, G96SSS (Milbert and Smith 1996). As such,
Figure 4
(152K) shows the difference between the geoid undulations of G96SSS and the
undulations of EGM96, corrected for the height anomaly to geoid undulation (Rapp 1996). The
magnitudes of this figure range from -186 cm in magenta to +334 cm in red. In this figure only,
we use the more recent gravity data sets and theory to show our final version of potential
"commission error" which is inherent in EGM96. This figure is not completely consistent with
Figure 6 of the beta report
, but is very useful in showing the areas where surface data and
EGM96 differ from one another.
The behavior of the geoid in the Pacific Northwest is of special interest due to its rugged
terrain and difficulty in matching GPS benchmarks with the geoid in the area. As per the beta
model test, a study of the EGM96 model in the area of 38-49 N and 230-255 E was made. In
that area, 505 GPS benchmarks are found. To avoid redundancy with the beta model report, only
the EGM-X05 results are repeated below, along with the EGM96 results. The statistics are
ordered by ascending elevation groups. Additionally, first, second and third differences are
presented.
Table 2 shows that EGM96, as mentioned earlier, is something of an outlier relative to
the five beta models (although, to be succinct we did not repeat the results of X01-X04). While
its 2nd and 3rd differences have similarities to EGM-X05, the differences in offsets at various
elevation groups show EGM96 to lie outside the three families (X01, X02 and X03-X05)
identified in the beta model report. Table 2 corresponds with Table 5 in the beta model report.
Results in Table 2, as well as the information contained in
Figure 4
(152K), show that EGM96 also
contains the elevation dependent commission error evident in the earlier beta models. Additional
investigations with subsets of the GPS benchmark residuals (not reproduced here) show the
elevation dependence is not due to GPS ellipsoid height scatter above 42 degrees latitude
(discussed in the next section).
X05 Model Cohort n RMS(cm) Offset(cm.) Diff 2nd-Diff 3rd-Diff
0- 500 138 49.27 71.02
500-1000 97 31.76 69.20
1000-1500 93 28.49 42.93 -26.27
1500-2000 91 32.98 34.90 -8.03 18.24
2000-2500 71 29.93 18.54 -16.36 -8.33 -26.57
Over 2500 15 35.05 -12.43
----------------------------------
All 505 43.14 49.13
EGM96 Model Cohort n RMS(cm) Offset(cm.) Diff 2nd-Diff 3rd-Diff
0- 500 138 45.68 68.07
500-1000 97 31.45 66.24
1000-1500 93 29.16 44.42 -21.83
1500-2000 91 33.49 39.00 -5.42 16.41
2000-2500 71 28.69 25.18 -13.82 -8.41 -24.81
Over 2500 15 38.22 -3.46
----------------------------------
All 505 40.21 49.97
High Resolution Geoid Computation and Evaluation
The details surrounding the computation of high resolution geoid models are found in the
beta model report. Using the identical procedures and data sets as the beta models, a new high
resolution geoid model for the United States was produced, which we call model 9696 (Please
note that this is not the same as G96SSS, as the methods and data sets used in the beta model
report are somewhat inconsistent with the final data and procedures of G96SSS). Model 9696 is
consistent with models 9620 - 9624 (see beta report), which are computed using EGM-X01 -
EGM-X05 respectively.
In a way similar to the low resolution model test, a tilted plane was fit to the high
resolution models, to find how additional gravity data can be used to remove commission error in
the low resolution models. The results were encouraging, showing that most of the tilt which
appears in the low resolution models is removable by computing a high resolution model. The
statistics of the fitted planes are found in Table 3 below, which corresponds to Table 2 of the beta
report.
Model Offset(cm.) Tilt(ppm) RMS about plane(cm) Azimuth(deg) 9620 -4.58 0.04 35.07 293 9621 -5.90 0.06 36.14 317 9622 -2.07 0.05 36.14 319 9623 -2.05 0.06 36.56 321 9624 -2.05 0.05 36.52 321 9696 -1.99 0.02 34.63 304
Table 3 shows an interesting, and somewhat surprising result. Considering the larger
magnitude of the tilt of EGM96, compared with the beta models, it is surprising that the final
impact on the high resolution geoid undulation grid would be to show a slight improvement. As
will be mentioned in a later section, the ability to remove commission error through
high-resolution geoid computation does not include the ability to remove any form of very long
wavelength (i.e. tilt) errors in the models. As such, we are pleased for the reduction in the
magnitude of tilt of a high resolution geoid, due to the use of EGM96.
Care should be taken, however, in interpreting the small magnitude of the tilt in the high
resolution models. Scatter plots of the residual geoid undulations show distinct problems in
certain areas, which happen to have the property of canceling one another in their contribution to
a continental tilt. These scatter plots are presented in the next 3 figures.
Figure 5
shows the
geoid residuals relative to orthometric height. As we expect, there is significantly less scatter
than in
Figure 1
, and a dependence of geoid undulation residual upon height does not appear to
be present.
Figure 6
shows the scatter relative to latitude, and
Figure 7
shows it relative to
longitude. In all three figures, certain characteristic problems are noted. First, a low-elevation,
south and east cluster of negative residuals is part of a difficulty in modeling the Florida geoid (a
problem that has subsequently been corrected for G96SSS). Also, a mid/high elevation, north
and west spread of residuals represents the difficulties in modeling the Pacific northwest (again,
some of this spread has tightened up with G96SSS). These features happen to have canceling
effects upon the tilt of the high-resolution model (i.e. a least-squares adjustment fits a flat line
through
Figure 6
, even though the general pattern is curved).
To better get an understanding of any potential elevation dependence that may not be
immediately obvious in
Figure 5
, an "elevation cohort" test, identical with that of the beta model
report, was conducted. In this test, geoid undulations residuals were grouped into elevation
groups, where benchmarks near Florida (south of 30) and Washington State
(north of 48) are
withheld from the analysis. The results are presented in Table 4 below. Table 4 corresponds
with Table 3 of the beta model report.
9624 Model Cutoff(m) n RMS(cm) Offset(cm.) Diff 2nd-Diff
(X05) 0 2190 24.73 9.84
1000 448 23.83 6.97
1500 268 24.80 6.88 -0.09
2000 107 24.50 7.43 0.55 0.64
9696 Model Cutoff(m) n RMS(cm) Offset(cm.) Diff 2nd-Diff
(EGM96) 0 2190 25.15 8.52
1000 448 24.52 3.22
1500 268 25.43 3.16 -0.06
2000 107 25.17 3.58 0.42 0.36
Table 4 enables us to discern more about the potential for elevation dependence in the
geoid undulation residuals for the high resolution models. Previously, model 9624 (based on
EGM-X05) showed the most uniform average offsets (as seen in the magnitudes of 1st and 2nd
differences). However, model 9696 (based on EGM96) shows even greater stability in the
average offsets. On the other hand, a larger RMS and greater discrepancies between average
offset of all GPS benchmarks and those above 1000 m shows a greater dependence on elevation,
and thus could be considered an artifact of a tilt in the model.
Comparison of EGM96 with EGM-X05
Prior to the release of EGM96, our preference amongst the beta models was EGM-X05.
This was the model used in the final preparations and tests for G96SSS/GEOID96, in
anticipation of the final EGM96 release. As such, it was interesting to compare these two models
with one another, and see what was the final effect of using EGM96 rather than EGM-X05 in
high resolution geoid calculations. Geoid undulations were evaluated directly on the geoid
(rather than using an up/down procedure) for both EGM-X05 and EGM96, with the differences
(EGM96 minus EGM-X05) shown in
Figure 8
(95K), covering the range of 24-53 degrees North in
latitude, and 230-294 degrees East in longitude. This image, done in color to aid in clarity,
shows large bowl shaped discrepancies, with a peak-to-peak magnitude of 95 cm (magenta
values are -49 cm, red values are +46 cm). Reports from the EGM96 team indicate that the most
significant change between EGM-X05 and EGM96 occurred in the satellite (i.e. degree 2 to 70)
solution. This agrees with the most pronounced undulation differences, which seem to have a
characteristic wavelength of about 8-10 degrees (around spherical harmonic degree 40).
To confirm our visual findings, radially averaged power was computed for the region
25-50 N, 258-283 E on the differences seen in
Figure 8
(95K). A plot of the power against the
frequency is shown in
Figure 9
on a log scale, and in
Figure 10
on a log-log scale. The spike at
the left of
Figure 9
is indicative of additional power at a specific frequency band. To help isolate
the frequency band,
Figure 10
is plotted on a log-log scale. In this case, any deviation from a
linear graph indicates a signal that is specific to one particular scale. In this case, we see the
spike of
Figure 9
translates into a large bump, deviating from the linear trend, around 0.1 - 0.2
cycles/degree, or a wavelength of about 5-10 degrees. This agrees nicely with the visual estimate
obtained from
Figure 8
. A signal of this wavelength corresponds roughly to spherical harmonic
degrees 30-50, which is within the spectral range of the satellite only, and combination solutions
of EGM96 (to degree 70).
In an effort to determine whether these large undulating differences were corrections, we
investigated one of the largest differences between EGM-X05 and EGM96: the 1 ppm tilt across
Oklahoma. As seen in
Figure 8
(95K), this tilt connects a low (blue) in Texas to a high (red) in Kansas.
Figure 11
shows a contour plot of the tilt, and the position of 16 GPS/level benchmarks in the
area (indicated by "+"). Two tests against "ground truth" were made. In the first test, surface
gravity data were gridded (see Smith and Milbert 1997) and the anomalies implied by both
EGM-X05 and EGM96 were removed. These two 2' x 2' residual gravity anomaly grids were
run through the 1-D spherical FFT form of Stokes' integral to generate two 2' x 2' residual geoid
undulation grids for the United States (with 8056 points in the study area). The size of the
residual undulations is indicative of the fit between the surface gravity data and the two models
(a comparison of the residual free-air anomalies could also be made, but we were interested in
the fit of undulations at this point). Table 5 shows the results of this comparison.
| Model name |
Average residual undulation |
RMS about the average |
| EGM-X05 |
13.1 cm |
12.9 cm |
| EGM96 |
18.5 cm |
16.6 cm |
Table 5. Residual geoid undulations in Oklahoma
Table 5 shows a clear indication that for this area, the 1 ppm slope induced by changing
from EGM-X05 to EGM96 is introducing a disagreement with the geoid implied by surface
gravity. To further investigate this finding, a second test was made. In the second test, the geoid
undulations implied by the 16 GPS/level benchmarks in Oklahoma were compared with the
undulations computed solely from the n=360 models. The orthometric heights were originally
NAVD 88 values, but were corrected for a 43.4 cm bias in NAVD 88. As in the "low resolution
geoid" section of this report , the undulations implied by the two models were corrected from
surface height anomalies to geoid undulations. The undulations from the two models were
subtracted from those implied by the GPS/level benchmarks, and the results of this comparison
are shown in Table 6 below.
|
Model Name |
Average difference |
RMS about the average |
| EGM-X05 |
-9.1 cm |
13.3 cm |
| EGM96 |
-17.5 cm |
14.9 cm |
Table 6. Residual geoid undulations on GPS/level benchmarks in Oklahoma
Table 6 confirms the findings of Table 5, that the features introduced in Oklahoma by
EGM96 are causing larger disagreements with surface data. It must be noted, however, that due
to the semi-periodic nature of the differences between EGM96 and EGM-X05, a number of such
slopes are induced across the United States, but time prevented us from studying them further.
Oklahoma was chosen as an area with good gravity and GPS on benchmarks coverage, as well as
coinciding with one of the larger differences between the models. Further studies should be done
to determine the magnitude of the other differences in their respective regions.
As previously mentioned, much of the commission error in the spherical harmonic
models can be removed by the use of the remove-compute-restore procedure in the creation of a
high resolution model. However, any commission error with a wavelength longer then 29
degrees (i.e. spherical harmonic degrees 0 through 6 or 7) will not be repairable, due to the
limited size (29 degrees north/south) of the grid which covers the United States. To exemplify
this point, two high resolution geoids were produced using our most recent data sets, and
procedures: one geoid based on EGM-X05, and one based on EGM96 (this second geoid is what
we've termed G96SSS). The differences between the two geoids has been calculated, and a plot
(minus a few degrees along the edges to reduce the visual impact of edge effects) of the
differences is shown in
Figure 12
. Aside from the edge effects, the difference between the geoids
is seen to be a very long wavelength structure across the United States. There is no indication of
the large bowl-shaped differences seen in
Figure 8
(95K). Thus, it seems clear that the structure of the
geoid is not significantly dependent upon the shorter (29 degrees or smaller) wavelengths of the
spherical harmonic model, but is rather controlled by the gravity data themselves. Any long
wavelength (29 degrees or greater) errors, however, that may be in the model are not removable
using this procedure.
Conclusions
Upon first inspecting the differences between EGM96 and EGM-X05, it was surprising to
see how large, and especially how semi-periodic, the differences were. These differences were
primarily seen as bowl-shaped features approximately 1000 km wide, superimposed over a slight
tilt. Overall, however, no significant change in the continental GPS/BM residual statistics
occurred with EGM96, aside from the larger tilt. A regional study shows that one of the larger
differences between EGM96 and EGM-X05 causes disagreement with surface data, but these
regional features are removed in the remove-compute-restore procedures used in high-resolution
geoid modeling. The importance of EGM96 became evident when it was used in the creation of
model 9696, a high resolution model that was less tilted, and slightly more consistent at high
elevations, relative to GPS benchmarks. This property, coupled with the known global
improvements of EGM96, make us confident that EGM96 will serve as a firm foundation for the
G96SSS and GEOID96 high-resolution geoid height models.
Acknowledgments
We would like to extend our sincere thanks and congratulations to all persons involved in
the production and dissemination of EGM96. In addition, the coordinating efforts of Michael
Sideris allowed an outstanding availability of global results from all members of the SWG. The
efforts of numerous NGS employees in the creation and evaluation of the gravity, NAVD88 and
GPS data were essential to this effort. The
National Imagery and Mapping Agency
(NIMA, formerly DMA) provided a major portion of the NGS land gravity data, and was instrumental in
the creation of various 3" and 30" digital elevation data grids in use. Dr. Walter Smith, NOAA,
provided the
altimeter-derived gravity anomalies.
References
Milbert, D.G. and D.A. Smith, 1996: Converting GPS Height into NAVD 88 Elevation with the GEOID96 Geoid Height Model . Proceedings of GIS/LIS '96 Annual Conference and Exposition, Denver, November 19-21, 1996, American Congress on Surveying and Mapping, Washington, D.C., pp. 681-692.
Rapp, R.H., 1996: Use of potential coefficient models for geoid undulation determinations using
a spherical harmonic representation of the height anomaly/geoid undulation difference.
Submitted to Journal of Geodesy.
Smith, D.A. and D.G. Milbert, 1997:
Evaluation of Preliminary Models of the Geopotential in the United States
, IGeS Bulletin, International Geoid Service, Milan, Italy, in press.
Figure Captions:
Figure 1 : Residual geoid undulations, GPS/Benchmarks vs. EGM96, relative to height
Figure 2 : Residual geoid undulations, GPS/Benchmarks vs. EGM96, relative to latitude
Figure 3 : Residual geoid undulations, GPS/Benchmarks vs. EGM96, relative to longitude
Figure 4 : Geoid Undulation Residuals, G96SSS vs. EGM96. (152K)
Figure 5 : Residual geoid undulations, GPS/Benchmarks vs. 9696, relative to elevation
Figure 6 : Residual geoid undulations, GPS/Benchmarks vs. 9696, relative to latitude
Figure 7 : Residual geoid undulations, GPS/Benchmarks vs. 9696, relative to longitude
Figure 8 : Geoid undulation differences, EGM96 minus EGM-X05. (95K)
Figure 9 : Radial power of EGM96 minus EGM-X05 undulation differences
Figure 10 : Radial power of EGM96 minus EGM-X05 undulation differences
Figure 11 : Undulation differences in Oklahoma, EGM96 minus EGM-X05, with GPS Benchmarks
Figure 12
: High-resolution undulation differences, CI = 1 cm
