COMPUTING NAD 83 COORDINATES

USING ITRF-DERIVED VECTOR COMPONENTS

Tomás Soler

Neil D. Weston

Heeyul Han

National Geodetic Survey, NOS, NOAA

1315 East-West Highway

Silver Spring, MD 20910, USA

[**Tom.Soler@noaa.gov**](mailto:Tom.Soler@noaa.gov)

[**Neil.D.Weston@noaa.gov**](mailto:Neil.Weston@noaa.gov)

Heeyul.Han@noaa.gov

ABSTRACT

Recently, a new set of 14 transformation parameters was jointly adopted
by National Resources Canada (NRCan) and the U.S. National Geodetic
Survey (NGS) for transforming coordinates between the International
Terrestrial Reference Frame of 2000 (ITRF00) and the North American
Datum of 1983 (NAD 83). Although a unique set of similarity
transformation equations could be applied everywhere to transform
positional coordinates between these two terrestrial frames, a more
consistent set of NAD 83 coordinates can be obtained if, locally, ITRF00
intersite vectors are transformed and constrained in the NAD 83 frame.
This is the procedure implemented by the OPUS (On-line Positioning User
Service) software, available through the NGS Tool Kit web page, to
derive positional coordinates on the NAD 83 datum. In this particular
case, the final NAD 83 coordinates of the unknown point are computed
from three GPS-derived intersite vectors radiating to nearby CORS
(Continuously Operating Reference Station) sites.

INTRODUCTION

A significant breakthrough in the fields of geodesy and surveying was
accomplished with the introduction of the technology and methods of the
Global Positioning System (GPS). Credit should go to GPS for
substantially increasing our knowledge of geocentric terrestrial
coordinate systems. It represents a giant leap from outdated classical
datums (North American Datum of 1927: NAD 27; South American Datum of
1956: SAD 56, etc.) and the TRANSIT predecesors. The absolute accuracies
currently available in geocentricity, orientation, and scale of
terrestrial coordinate frames will be difficult to improve significantly
and should last well into the next decade. This achievement has been
possible thanks to refinements in post-processed precise ephemeris which
today provide the orbital positions of the GPS satellites at the 5 cm
level.

As a result of this and other improvements, we have entered into a new
specialized realm of science that is purely geometrical in concept and
strictly coordinate-centered. A large amount of effort has been expended
to exploit the advent of precise positioning. Several countries have
readily deployed permanent GPS satellite tracking networks to provide
accurate referencing for differential GPS positioning techniques in
order to maintain geodetic control at sub-centimeter accuracies. GPS
campaigns have proliferated in recent years and new superior continental
datums have evolved around the globe (e.g., European Terrestrial
Reference Frame of 1989: ETRF89; Geocentric Datum of Australia 1994:
GDA94; NAD 83(CORS96); South American Geocentric Reference System:
SIRGAS, etc.). Subsequently, local networks have been densified and
rigorous control for mapping and surveying extended. Also, as a result
of improved national geoid height models --so critical for GPS
leveling---GPS now affords a more economical alternative to differential
leveling for many heighting applications.

GPS users can now determine the three-dimensional coordinates of a point
with centimeter accuracy relative to a control station located several
hundred kilometers away (Eckl et al., 2001). As mentioned above, there
is a proliferation of continuously operating reference stations (CORS)
that can be operated for a myriad of scientific applications. Such is
the case of the National Geodetic Survey (NGS) network named National
CORS, at present --January, 2002\--, a set of 253 permanently monumented
GPS stations whose coordinates define the National Spatial Reference
System (NSRS). The National CORS network is becoming not only a natural
tool for accurate 3D geodetic positioning in the United States, but its
GPS data holdings are used by a plethora of investigators interested in
ionospheric research, crustal motions, water vapor studies, etc. (Snay
and Weston, 1999). For more information about CORS, consult the web
address http://www.ngs.noaa.gov/CORS/. The NSRS is based on the North
American Datum of 1983, NAD 83 for short.

Coordinates of a CORS station are determined by NGS by exploiting the
most advanced GPS methodologies and software. These points, initially,
are rigorously known in a well-defined coordinate frame, such as the
International Terrestrial Reference Frame of 2000 (ITRF00). Introductory
information about ITRF frames can be found in Boucher and Altamimi
(1996) and Snay and Soler (2000b). The final step is to transform ITRF00
(epoch 1997.0) coordinates into NAD 83 coordinates. This requires a set
of transformation equations that will be explained in the following
section. A brief description of the evolution of NAD 83 is given in Snay
and Soler (2000a) where the concept of frame "realization" is explained.
For example, the latest realization of NAD 83 based on GPS observations
is denoted NAD 83 (CORS96), although to simplify writing, this
nomenclature will be truncated in this article to NAD 83.

TRANSFORMATION OF COORDINATES

In order to know the relationship between geocentric ITRF00 (1997.0)
coordinates and NAD 83 coordinates, a 14-parameter similarity
transformation between the two frames was derived. Not too long ago, the
Helmert (similarity) transformation between two independent sets of
coordinates, referred to two cartesian 3D frames, was defined as a
function of only seven parameters. However, present improvements in GPS
accuracies are able to detect changes as a function of time in the
customary seven parameters. In another words, today geodesists are not
satisfied in knowing only the typical three translations, three
rotations, and the scale differential factor connecting any two 3D
frames, but, for highly accurate work, the rate of change of these
parameters with time is also required. Consequently, the general
formulation to transform coordinates between two frames, say ITRF00 to
NAD 83, is given according to the following equations :

![](media/image1.wmf)

where for an adopted time of reference, denoted *t~0~*:

![](media/image2.wmf)

![](media/image3.wmf)

![](media/image4.wmf)

![](media/image5.wmf)4.8481368110^-9^ conversion factor from
milli-arcseconds (mas) to radians

In the above expressions:

![](media/image6.wmf)Shifts of the origin of the NAD 83 frame on the
geocentric ITRF00 frame at epoch ![](media/image7.wmf)

![](media/image8.wmf)Differential anticlockwise (counterclockwise)
rotations (expressed in radians) at epoch ![](media/image9.wmf),
respectively, around the ![](media/image10.wmf)axes to establish
parallelism with the ![](media/image11.wmf) axes.

![](media/image12.wmf)Differential scale change (expressed in ppb10^-9^;
ppb parts per billion)

![](media/image13.wmf)Variation of the shifts with respect to time,
e.g., ![](media/image14.wmf)

![](media/image15.wmf)Variation of the rotations with respect to time,
e.g., ![](media/image16.wmf)

![](media/image17.wmf) Variation of the scale factor with respect to
time, ![](media/image18.wmf)

Originally, coordinates at 12 points in Canada and the United States,
known on the ITRF96 and NAD 83, were used to determine the
transformation parameters (Craymer et al., 2000). Later updates by the
International Earth Rotation Service (IERS) provided us with the
required information to complete a transformation according to the
following mapping:

![](media/image19.wmf)

The final values of the 14 transformation parameters adopted by National
Resources Canada (NRCan) and NGS to transform coordinates given on the
frame ITRF00 to the geodetic datum frame NAD 83 are:

*t~0~* = 1997.0

Shifts: ![](media/image20.wmf) 0.9956 m ; ![](media/image21.wmf) 1.9013
m ; ![](media/image22.wmf) 0.5215 m

Rotations: ![](media/image23.wmf) 25.915 mas ; ![](media/image24.wmf)
9.426 mas ; ![](media/image25.wmf) 11.599 mas (mas milli-arcseconds)

Scale factor: ![](media/image26.wmf) 0.62 10^-9^ (unitless)

Shift rates: ![](media/image27.wmf) 0.0007 m year ^-1^ ;
![](media/image28.wmf)0.0007 m year ^-1^ ; ![](media/image29.wmf) 0.0005
m year ^-1^

Rotation rates: ![](media/image30.wmf) 0.067 mas year ^--1^ ;
![](media/image31.wmf) 0.757 mas year ^--1^ ; ![](media/image32.wmf)
0.051 mas year ^--1^

Scale factor rate: ![](media/image33.wmf) 0.18 10^-9^ year ^--1^

The above set of equations are the ones implemented by NGS interactive
program HTDP (Horizontal Time-Dependent Position). This software is
without cost, available at the NGS web site
([**http://www.ngs.noaa.gov**](http://www.ngs.noaa.gov/)) by clicking on
«Geodetic Tool Kit», and then on «HTDP». Further technical information
about HTDP can be found in (Snay, 1999).

NGS recognized the implications and promise of GPS in its early stages
of development and embarked on the promotion and adaptation of GPS
methods to improve the NSRS. However this is an iterative process marked
by the constant demand of improving positional accuracies based on the
latest advancements on technology and updated definitions of coordinate
systems. A history of various sets of transformation parameters between
several realizations of ITRF and NAD 83 is available on NGS web page
([**http://www.ngs.noaa.gov/CORS/
metadata1)**](http://www.ngs.gov/CORS/metadata1)).

There is another consideration that should be kept in mind; the
transformation parameters necessary to obtain NAD 83 coordinates from
ITRF00 are the result of comparing coordinates at 12 points. However,
when geodetic networks such as the Federal Base Network (FBN) are
adjusted, some observation errors are propagated and small distortions
appear when coordinates obtained using ITRF00 and the strict application
of equation (1) are contrasted with the results of the FBN adjustment.
These small differences (up to 2 cm in the horizontal components and 4
cm on the vertical) are also influenced by the fact that FBN network
results are an ensemble of discrete state networks that were developed
and adjusted independently although constrained to CORS. For this reason
a continent-wide adjustment including all GPS observations stored in the
NGS data base, is planned to by published by 2005. In the mean time,
when no adjustments are performed, the procedure of using observed
intersite vector components transformed to the NAD 83 was adopted as
described in the next section.

TRANSFORMATION OF VECTOR COMPONENTS

Recall that, initially, GPS processing determines the coordinates of the
unknown points at the epoch of observation *t* referred to the frame
orientation implicitly defined by the ephemeris used during the
processing stage. NGS usually relies on the precise ephemeris
disseminated by the International GPS Service (IGS) which, presently,
assures an orbit of highest accuracy. Generally, the final
post-processed products available to the GPS user are vector components,
that is, the difference between the determined coordinates of the
unknown point P and the coordinates of the fixed (base; fiducial) point
A at the time of observation *t*. The magnitude (modulus) of the vector
![](media/image34.wmf)is the baseline length *AP*. Mathematically, in
matrix notation, we can write the vector of the three components in
either of the following forms:

![](media/image35.wmf)

NGS typically stores processed vectors in its data base in some ITRF
frame. Similarly, people familiar with NGS' ADJUST software know that
this is the format required by the input G-file which contains all
vector components of each session referred to a topocentric frame
parallel to ITRFxx, located at the origin of the vector. The file also
includes the standard error of each vector component and their
correlation matrix. General matrix equations to rigorously transform
vector components and their variance-covariance matrix at different
epochs are given in Soler (2001).

Applying the theory described above, it is possible to transform the
observed components at time *t* from ITRF00 to NAD 83. Notice that
because the vector components defined by equation (2) involve the
subtraction of two sets of coordinates, the shifts and shift rates in
equation (1) will cancel. Substituting equation (1) into (2) twice, we
arrive at the equation to transform ITRF00 components into NAD 83
components:

![](media/image36.wmf)

where the values of the parameters required in the above equation were
given previously. Equation (3), with the NAD 83 coordinates of the CORS
stations, is used internally by NGS software OPUS (On-line Processing
User Service; http://www.ngs.noaa.gov/OPUS/index.html) to obtain the
final NAD 83 coordinates of the unknown point P. Thus, OPUS computes the
coordinates of the unknown point P using the components of the vectors
radiating to the three closest CORS stations.

The logic applied in OPUS is as follows. Assume that we want to know the
coordinates of an arbitrary point P. Using IGS ephemeris and the RINEX
data at three nearby CORS sites (name them A, B, and C) whose
coordinates on the ITRF00 are known at the epoch of observation *t*,
OPUS determines the coordinates of P at time *t* on the frame ITRF00.

These values are also given to the OPUS user on the output. However,
most surveyors working on the United States need coordinates on the NAD
83. At this point, OPUS could have used program HTDP to convert ITRF00
coordinates into NAD 83 coordinates, however this approach would have
assumed that the adopted 14 transformation parameters between ITRF00 and
NAD 83 fit equally well in all areas of the country. This may not be the
case due to the effect of small localized distortions. Thus, it was
decided that instead of applying a general 14 parameter transformation
using equations of the type (1), the propagation from the NAD 83
coordinates at the CORS stations A, B, and C, using actually observed
vector components would be more realistic for the calculation of the NAD
83 coordinates at new unknown points P. It is at this stage that the
components of the vectors are introduced.

Initially, the three components on the ITRF00 frame at time *t* of the
three vectors ![](media/image37.wmf), and ![](media/image38.wmf) are
computed: ![](media/image39.wmf), ![](media/image40.wmf), and
![](media/image41.wmf) where the superscript *T* stands for matrix
transpose. The epoch of observation *t* is always given on the OPUS
output (e.g., t = 2002.7696). Substituting these values in equations (3)
after introducing the 14 transformation parameters described before, one
obtains the components of the vectors ![](media/image37.wmf), and
![](media/image38.wmf) referred to the NAD 83 frame. Finally the adopted
NAD 83 coordinates of the CORS stations (points A, B, and C) are invoked
to determine the coordinates of the unknown point P also referred to the
NAD 83:

![](media/image42.wmf)

Clearly, two similar equations could be obtained using the coordinates
and vector components at the other two CORS stations B and C.
Consequently, one ends up with three independent determinations of the
unknown point P now referred to the NAD 83 frame. The final value given
by the OPUS output is the average of the three independent
determinations. As a measure of precision, OPUS prints the peak--to-peak
error (maximum difference among the three determined values) for each
component in linear units. Other ancillary information e.g. orthometric
height using GEOID99 and NAVD 88, UTM northing and easting, baseline
lengths, etc., are given.

CONCLUSIONS

The theory described above is the one applied by the OPUS software to
compute NAD 83 coordinates. The program determines the coordinates of an
arbitrary point P requiring as input the observation RINEX file (2 hours
duration is the minimum allowed), the antenna part number (this is
important because the software implements antenna phase pattern
corrections), and the height of the ARP (antenna reference point) above
the point of interest. The hands-off software uses the closest three
CORS sites with good data to compute the coordinates of P. Depending on
data quality, some of the CORS stations may be replaced by other nearby
sites. Recently it was proved (Eckl et al., 2001) that baseline length
is not critical to obtain accurate GPS-derived coordinates,
consequently, the distance of the unknown P to the CORS stations does
not have an important effect on the results and priority is given to the
quality of the data to obtain accurate positioning.

AKNOWLEDGMENTS

The authors are indebted to Dennis Milbert, NGS, who suggested the use
of vector components to determine NAD 83 coordinates in OPUS, and
Richard Snay, also at NGS, for helpful discussions and a thorough review
of an early version of the manuscript.

REFERENCES

Boucher, C. and Z. Altamimi (1996). International Terrestrial Reference
Frame, *GPS World*, 7(9): 71-74.

Craymer, M., R. Ferland, and R. Snay (2000). Realization and unification
of NAD 83 in Canada and the U.S. via the

> ITRF. In: R. Rummel, H. Drewes, W. Bosch, H. Hornik (eds), *Towards an
> Integrated Global Geodetic Observing System (IGGOS)*, IAG Section II
> Symposium, Munich, October 5-9, 1998, International Association of
> Geodesy Symposia, Volume 120: 118-121, Springer-Verlag, Berlin.

Eckl, M.C., R.A. Snay, T. Soler, M.W. Cline, and G.L. Mader (2001).
Accuracy of GPS-derived relative positions

as a function of interstation distance and observing-session duration,
*Journal of Geodesy*, 75(12): 633-640.

Snay, R.A. (1999). Using the HTDP software to transform spatial
coordinates across time and between reference

frames, *Surveying and Land Information Systems*, 59(1): 15-25.

Snay, R.A. and Soler, T. (2000a). Modern Terrestrial Reference Systems.
Part 2: The evolution of the NAD 83,

*Professional Surveyor*, 20(2): 1-2.

Snay, R.A. and Soler, T. (2000b). Modern Terrestrial Reference Systems.
Part 3: WGS84 and ITRS, *Professional*

*Surveyor*, 20(3): 1-3.

Snay, R.A., and N.D. Weston (1999). Future directions of the National
CORS System. In: *Proceedings 55^th^ of the*

*Annual Meeting of the Institute of Navigation*, June 28-30, Cambridge,
MA: 301-305.

Soler, T. (2001). Densifying 3D networks by accurate transformation of
GPS-determined vector components, *GPS*

*Solutions*, 4(3): 27-33.