**Recent Advances in the Acquisition and Use of Terrain Data for Geoid
Modelling over the United States**

D.A. Smith, D.R. Roman

NOAA/National Geodetic Survey

1315 East-West Highway, SSMC3, N/NGS5, Silver Spring, MD, 20910, USA

# 

# Abstract

# For much of the past decade, high resolution geoid models have been produced at the National Geodetic Survey (NGS). The use of digital elevation models (DEMs) has been important to the computation of geoid models, but has also been one of the primary limitations of the geoid accuracy due to both poor data quality and a series of computational approximations. Recent steps have been taken at NGS to replace existing DEM data with data of finer resolution and higher quality. Additionally, new computational tools are being implemented which no longer rely on accuracy-reducing approximations. By moving toward more accurately computed terrain-induced gravity signals, errors in gravity anomalies exceeding tens of mGals have been removed, and subsequently centimeters of geoid accuracy have been gained.

# 1 

# Introduction

The need for accurate terrain data in the computation of an accurate
geoid model has long been known. Whether one uses the Stokes-Helmert
approach or alternative methods, such as that of Molodensky, there has
always been a need for an accurate digital elevation model (DEM).
Although DEMs with horizontal resolution as fine as three arcseconds
have been available since the 1960's, their general accuracy has been
limited due to their data sources (small-scale digitized cartographic
maps) and lack of significant photogrammetric coverage. In addition to
DEM accuracy, the use of a DEM in computing geoid models is further
limited by computational resources (RAM, CPU speed, disk space) and
computational shortcuts (planar approximations, flat-topped prisms,
constant density, Faye anomalies approximating Helmert anomalies). At
NGS, even as late as 1996, the gravitational terrain signal was modelled
as planar, 2-D FFT terrain corrections at a horizontal resolution of 30
arcseconds (Smith and Milbert, 1999). Initial tests at NGS indicate that
significant (10 cm and greater) errors were being made in the geoid
model due to the coarse resolution of the DEM, existing large-area
biases in the DEM, the computational approximations in computing terrain
corrections, and the use of Faye anomalies instead of Helmert anomalies.
Two separate efforts were begun at NGS in 1998 to improve this
situation. The first was a search for a more accurate DEM for the United
States, and the second was taking some initial steps toward removing
classical approximation in terrain corrections. Both were successful,
and may be implemented in the GEOID99 model, anticipated for release in
September 1999. (http://www.ngs.noaa.gov/GEOID/GEOID99/geoid99.html)

# 2 

# Testing New Digital Elevation Models

In the United States, there are only two significant sources of high
resolution (better than 30 arcseconds) DEMs. The first is 7.5 minute
quadrangle DEM from the United States Geological Survey (USGS) and the
other is Digital Terrain Elevation Data (DTED) Level 1 from the National
Imagery and Mapping Agency (NIMA). The comparison between the two
available DEMs is made in Table 1.

It was clear from the initial comparison that the USGS data was somewhat
superior to the NIMA data, but in order to be sure, two tests were
conducted on these data: 1) Agreement with point heights and 2) Terrain
correction tests.

**Table 1.** Comparison of USGS and NIMA high resolution Digital
Elevation Models

+------------------+-----------------------+---------------------------+
| Category         | USGS                  |                           |
+------------------+-----------------------+---------------------------+
| DEM Name         | 7.5 minute quadrangle | DTED Level 1              |
|                  | DEM                   |                           |
+------------------+-----------------------+---------------------------+
| Horizontal       | 30 meters             | 3 arcseconds              |
| Resolution       |                       |                           |
+------------------+-----------------------+---------------------------+
| Grid             | UTM                   | Geographic ( / )          |
+------------------+-----------------------+---------------------------+
| ![](me           | Free on WWW           | Proprietary               |
| dia/image1.wmf){ |                       |                           |
| width="2.8125in" |                       |                           |
| height="2.       |                       |                           |
| 1770833333333335 |                       |                           |
| in"}Availability |                       |                           |
+------------------+-----------------------+---------------------------+
| Format           | SDTS                  | NIMA DTED                 |
+------------------+-----------------------+---------------------------+
| Vertical Datum   | Well-defined (often   | Unknown (named "Mean Sea  |
|                  | NGVD 29)              | Level")                   |
+------------------+-----------------------+---------------------------+
| Horizontal Datum | Well-defined (often   | Unknown (WGS72? WGS84?)   |
|                  | NAD 27)               |                           |
+------------------+-----------------------+---------------------------+
| Sources          | 1:24,000 scale maps   | 1:250,000 scale maps and  |
|                  | and photogrammetry    | 25 other sources          |
+------------------+-----------------------+---------------------------+
| Biggest Problems | a)  1.1% of files     | a)  Does not always       |
|                  |     have blunders     |     contain true 3        |
|                  |                       |     arcsecond spectral    |
|                  | b)  Limited quality   |     information           |
|                  |     control           |                           |
|                  |                       | b)  Poorly identified     |
|                  | c)  No data outside   |     datum information     |
|                  |     the USA           |                           |
|                  |                       | c)  Some cells have       |
|                  | d)  Partially         |     biases over 10 meters |
|                  |     complete (85%)    |                           |
|                  |     over USA          |                           |
+------------------+-----------------------+---------------------------+

**2.1**

**DEM heights versus point heights**

The first test on the DEMs was to compare their gridded heights to point
heights stored in the NGS database. This is important because both
sources of heights are used in geoid modeling, and consistency is
therefore desirable. The point heights were predominantly scaled from
topographic maps by surveyors during gravity surveys. Although these
heights can be erroneous, the variety of surveyors who gathered such
heights supports the idea that in a large sample of point heights,
errors are expected to be more or less random. Two separate 1 x 1 degree
areas of the Pacific Northwest area of the USA were chosen for this
test, based on the ruggedness of the topography in that region. The
geographic limits of the two areas were \-- Area A: 47-48 N and 237-238
E, Area B: 46-47 N and 240-241 E. Area A had 1076 point heights in the
NGS database and Area B had 397 point heights. In each area, the gridded
DEM heights were biquadratically interpolated to the location of the
point heights. A difference between the heights was made, and histograms
of these differences made. In order to fairly compare the USGS DEM to
the NIMA DTED, the USGS DEM was decimated to 3 arcseconds first, and
then height comparisons were made. Figures 1 and 2 show the histograms
for Area A, and Figures 3 and 4 show them for Area B.

**Fig. 1** Mismatch between USGS 30 meter DEM and point heights in the
NGS Gravity database, Area A.

![](media/image2.wmf){width="2.8125in"
height="2.1770833333333335in"}**Fig. 2** Mismatch between NIMA Level 1
DTED and point heights in the NGS Gravity database, Area A.

![](media/image3.wmf){width="2.8125in"
height="2.1770833333333335in"}**Fig. 3** Mismatch between USGS 30 meter
DEM and point heights in the NGS Gravity database, Area B.

![](media/image4.wmf){width="2.8125in" height="2.1770833333333335in"}

**Fig. 4** Mismatch between NIMA Level 1 DTED and point heights in the
NGS Gravity database, Area B.

Although these figures only represent two areas, they already show a
clear indication that the USGS DEM data are superior to the NIMA DTED
Level 1. Note the tighter histograms of USGS (Figures 1 and 3), compared
to the more spread out ones of NIMA (Figures 2 and 4). This indicates a
smaller standard deviation of the differences between USGS DEM heights
and the point heights, and thus better agreement in general. It is also
indicative of the lack of true 3 arcsecond data contained in the NIMA
DTED Level 1 data. Additionally, note in Figure 4 the large (20 meter)
shift of the histogram mean from zero. This is evidence of a 20 meter
bias which exists in the NIMA DTED data of this region. Discussions with
NIMA personnel indicate that the source of this bias is currently
unknown. In general, therefore, it is seen that the USGS data seem to
have a better agreement with existing point heights than DTED Level 1.

**2.2**

**Terrain Corrections from DEMs and their impact on the geoid**

As a second test, terrain corrections (TCs) were computed from both
DEMs, and compared to existing TCs based on the existing 30 arcsecond
DEM, as well as compared to one another. The impacts of the various TCs
on the geoid were also computed and compared. Table 2 shows the
differences between various TC computations, and Table 3 shows the geoid
impact of the differences seen in Table 2. In Table 3, the "Critical
Distance" is the distance (in km) away from the maximum geoid impact, at
which the geoid impact still exceeds 1 cm.

**Table 2.** Differences between various TC computations, Area A.

  ------------------ ------------------------ ----------------------------
  Category           USGS minus 30"           USGS minus NIMA

  Average            1.4 mGals                0.7 mGals

  Maximum            20 mGals                 18mGals
  ------------------ ------------------------ ----------------------------

**Table 3.** Geoid impact of differences between various TC
computations, Area A.

  ------------------ ------------------------ ----------------------------
  Category           USGS minus 30"           USGS minus NIMA

  Maximum            9.4 cm                   5.8 cm

  Critical distance  400 km                   200 km
  ------------------ ------------------------ ----------------------------

Certain conclusions can be drawn from these tables, and are further
validated if one makes a detailed study of color plots (not shown in
this paper) of the various data that are the foundation of these tables.
First, it is clear that the larger average TCs of USGS relative to both
the existing 30 arcsecond TCs as well as the NIMA Level 1 DTED TCs
indicates that more "signal" or "roughness" exists in the USGS DEM. This
is further evidence that the NIMA data are not completely portraying 3
arcsecond information, even though they are distributed with that
horizontal resolution. Secondly, these results were over a 1 x 1 degree
area, but note that geoid impacts of switching from 30 arcsecond to 3
arcsecond data can yield geoid changes up to 400 km away from the center
of Area A (which is outside of Area A). Thus there is the potential for
geoid changes (at the 1+ cm level) to come not only from the data inside
a 1x1 degree area itself, but also from the 8 surrounding 1 x 1 degree
areas, causing constructive interference at the few decimeters level.

![](media/image5.png){width="2.8125in" height="2.1770833333333335in"}

Having conducted these comparisons, it was clear that the USGS DEM would
provide the best elevation data in the USA. Since these data are on a
UTM grid, were on outdated datums (NAD27 and NGVD29) and contain some
blunders, it was necessary to clean and re-grid the data, and transform
onto modern datums (NAD 83 and NAVD 88) before being used on a large
scale at NGS. Additionally, the incomplete (85% total coverage) nature
of the available data meant that some DTED Level 1 data would also need
to be used. During the Spring and Summer of 1999, NGS cleaned, combined,
and regridded the USGS and NIMA data (giving priority to USGS data) into
a new 1 arcsecond DEM covering the Northwest quarter of the United
States, called NGSDEM99 (Smith and Roman, 1999). There are plans to
expand this effort in 2000 and create a 1 arcsecond DEM for the entire
United States.

# 3 

# New Computational Methods

![](media/image6.wmf){width="2.8125in"
height="2.1770833333333335in"}Many classical approximations in the
computation of terrain-induced gravity signals are being re-evaluated at
NGS for their accuracy. A few are mentioned in this paper.

**3.1**

**Spherical versus rectangular DEM prisms**

When computing the gravitational attraction induced by the terrain, a
planar approximation is often used. This approximation ignores the
slight, but calculable, effects of the curvature of the Earth in two
ways. The first, and more obvious, is the far-field spherical effect,
which is the effect that the terrain dips below the horizon in a
spherical co-ordinate frame, but not in planar. The second, more subtle
effect, is that the shape of prisms is different in spherical versus
planar co-ordinates, and this slight shape change has a noticeable
impact nearest the station of interest. This is referred to as the
near-field spherical effect and is illustrated in Figure 5.

**Fig. 5** Geometric illustration of the differences in near-field
prisms between planar and spherical coordinates

From the simple geometry of the near-field prisms, an empirical
difference was computed between the gravitational attraction of a
rectangular prism and a spherical prism. This difference can be
translated as one part of the error in computing Helmert anomalies when
using the planar approximation. The empirical error seems to be almost
entirely based on the height, H, of the prism and not on it's horizontal
extent, . The empirical function is shown in Figure 6:

**Fig. 6** Empirical error in Helmert anomaly computation due to the
prism-shape error in planar coordinates

Although the Helmert anomaly error is relatively small, the effect is
systematic and should be applied to every gridded DEM location where a
TC is currently computed using planar co-ordinates. If this is done,
this effect has a noticeable effect on the geoid, achieving over 7 cm in
the Rocky Mountains of the United States. A map of this geoid impact is
shown in Figure 7:

![](media/image7.wmf){width="2.8125in" height="2.1770833333333335in"}

**Fig. 7** Geoid impact (cm) of empirical errors in Helmert anomaly
computation due to the prism-shape error in planar coordinates

**3.2**

**Rigorous space domain computations**

Because of the regularly gridded nature of most DEMs, computations done
with them (such as gravitational attraction), are naturally suited to
spectral methods such as FFTs (Schwarz, Sideris and Forsberg, 1990).
However, transformation of data into the spectral domain can cause
inaccuracies in computations, and therefore it is necessary to have
highly accurate space-domain computations as a "golden rule" against
which to check all future spectral computations. With this in mind, NGS
performed an accuracy analysis, computing the gravitational signal
induced by the removal of local terrain, and the restoration of that
terrain as a 'condensed layer' (i.e. "Helmertization" of the terrain).
In that study (Smith, Robertson and Milbert, 1999), it was found that
high resolution DEM data was not necessary for large areas. In fact, a
cut-off radius of 0.2 degrees was seen as acceptable in all forms of
terrain for an accuracy of 0.01 mGals. Between 0.2 degrees and 4.0
degrees, high resolution DEMs may effectively be replaced with coarser
DEMs (such as 30 arcseconds). Outside of 4 degrees, even coarser DEMs
may be used (5 arcminutes) to compute the remaining global signal
induced by topography.

# 4 

# Conclusions

The use of digital elevation models in the computation of geoid models
is a long-established necessity. However, recently available DEMs, and
the latest computer resources have made it possible, and worthwhile, to
take steps beyond coarse DEMs with classical computational
approximations. Two high-resolution DEMs, from USGS and NIMA were tested
against one another, and the more accurate USGS data was used as the
foundation for a new 1 arcsecond DEM (NGSDEM99) for the Northwest United
States (Smith and Roman, 1999). This new DEM will be used in the next
high-resolution geoid model for the United States, GEOID99 (scheduled
for release September 1999;
http://www.ngs.noaa.gov/GEOID/GEOID99/geoid99.html)

To supplement the new, accurate DEM data that is available, NGS has also
been studying better ways of computing the terrain-induced gravity
signal. Errors in old DEMs were only one reason why previous models were
not of the best accuracy. The other reason has been the use of planar
co-ordinates, and lack of rigorous checks on spectral computational
methods. Future geoid models will make use of the latest techniques
(Smith, Robertson and Milbert, 1999) to remove centimeters and even
decimeters of lost accuracy in the geoid.

References

Schwarz, K.P., M.G. Sideris and R. Forsberg (1990). The use of FFT
techniques in physical geodesy. *Geophysical Journal International*, Vol
100, pp. 485-514

Smith, D.A., D.G. Milbert (1999). The GEOID96 high resolution geoid
height model for the United States*, Journal of Geodesy*, Vol 73, No. 5,
pp 219-236.

Smith, D.A., D.S. Robertson, D.G. Milbert (1999). The gravitational
attraction of local crustal masses in spherial coordinates, submitted to
*Journal of Geodesy*.

Smith, D.A. and D.R. Roman (1999). A new high resolution DEM for the
Northwest United States, submitted to *Surveying and Land Information
Systems*.