In North America, positions on Earth are commonly expressed in terms of 3D terrestrial reference systems. Three of the most popular of these include the North American Datum of 1983 (NAD 83), the World Geodetic System of 1984 (WGS 84) used by GPS and the International Terrestrial Reference System (ITRS).

Each of these systems has similar definitions for their reference frames. All are right-handed Earth-centered, Earth-fixed coordinate frames with their origin at the Earth’s geocenter, the z axis approximately coinciding with the Earth’s axis of rotation, and the x axis passing through the meridian of zero longitude (i.e., the International Reference Meridian) at the equator.

Though defined similarly, their actual realizations differ because the authorities responsible for establishing them use different station networks and derive the station velocities using different measurement techniques. For example, the ITRS is realized by a worldwide network of several hundred stations whose movements along the Earth’s crust are measured by various space geodetic techniques, while the original realization of NAD 83 was determined by a similarly large network of stations measured primarily by Doppler observations from Transit satellites, and more recent realizations of WGS 84 rely entirely on GPS observations by a relatively small set of tracking stations.

To complicate matters, these reference systems are occasionally refined as the volume of space geodetic data increases and the accuracy of positioning technology improves, resulting in the existence of several slightly differing realizations of the same reference system.

The coordinate values of the same location in different reference frames may differ by several meters. Therefore, mathematical transformation is necessary to convert all positions into terms of the same reference system.

Why Time Matters

The Earth is a dynamic body. Its center of mass and axis of rotation change over time. Its surface also consists of a number of semirigid crustal plates that are constantly in motion relative to each other. A point on a plate may move several centimeters per year with respect to a given reference frame.

Where two plates meet, either in direct collision or by rubbing against each other, the crust near the area of contact becomes deformed over time. Along fault lines, the crust intermittently and abruptly shifts when major earthquakes occur.

In addition, part of the North American plate--mostly in Canada--that was once weighted down with mile-thick ice sheets during the previous ice age is still moving or rebounding upward several millimeters per year, while the southern portion of the plate is moving in a slightly downward direction.

Because of these movements, a position that was observed at one epoch or point in time may have a significantly different location within a reference frame at another epoch. Time-dependent positioning therefore becomes important for anyone who wants to use GPS to its submeter precision potential. For example, GPS observations are often expressed in some realization of WGS 84 (or, for WAAS-corrected data, ITRF00), while end users frequently want positional data in NAD 83, which is the official reference system of the United States and Canada. If the difference between reference frames is not accounted for, the coordinates of the GPS data may be several meters off their NAD 83 values.

When a reference system is refined by a new realization, as typically happens every few years, a user may want points expressed in previous realizations to be converted to the new one.

The NGS HTDP Utility

Since the early ‘90s, and in response to the increasing use of GPS in precision geodetic surveying, the National Geodetic Survey (NGS) developed and has maintained an interactive command-line utility called HTDP (horizontal time dependent positioning) to convert time-dependent positional data.

The current version of HTDP supports coordinate conversion between 14 unique reference frames and models crustal motion and over two dozen dislocation events in North America from the time of the 1906 San Francisco earthquake to more recent events such as the 2002 Denali Fault earthquake in Alaska.

Simple conversion between reference frames is performed using a seven-parameter, 3D conformal transform. However, because the relative orientations of the reference frames may change over time, the transform parameters themselves are time-dependent. To account for this characteristic, each of the seven parameters (i.e., three axis offsets, three rotation angles and a scale factor) are independently adjusted by a factor of the time difference between the observation epoch of the point and the realization epoch of the reference frame. Thus, 14 transformation parameters are actually required for the conversion.

If the coordinates of a position are to be updated to a different epoch--either forward or backward in time from its current date--the horizontal movement of the tectonic plate on which the position resides is taken into consideration along with the cumulative effect of any dislocation events that may have distorted the crust at that position. The former is modeled as a constant motion, while the latter are modeled as discrete, instantaneous shifts at specific epochs.

Continuous plate motion along the western coast of North America, where tectonic activity is greatest, is modeled by velocity grid files from which a point’s velocity is interpolated. In other modeled regions, the velocity is computed from platewide angular velocities.

Although reference frame conversions may be performed globally, changes in epoch can only be performed in those areas where crustal motion has been modeled. Such areas are currently limited to the approximate bounds of the North American, Pacific, Caribbean, Juan de Fuca, Cocos, and Mariana (Philippine) plates, which encompass land in North America, Central America and many of the islands in the Pacific Ocean.

Enhancing Functionality With Advanced Tools

Converting HTDP coordinates on GIS files presents a number of challenges. One problem is an inability in the commonly used data formats to store the date with the coordinate system definition in the file. Essentially, once a position has been corrected for a particular date, there is no way for another software package to recognize the date associated with the coordinates that comprise the map data. This limitation forces date corrections to be actively maintained by the GIS practitioner in the metadata. Whenever key components of coordinate data cannot be automatically stored and interpreted by the software, there is a chance for user error. Due to the highly precise and often very small shifts that come from the time-dependent corrections, it is not always easy to detect where such errors occur.

Another common problem working with time-based corrections is introducing data from older, outdated datum models that do not support time-based corrections. The issue of converting between a time-based coordinate and a non-time-based coordinate is that the conventional coordinate transformation parameters used (three parameters, seven parameters, etc.) are often much less accurate and precise than that of the HTDP transformation. Despite being able to convert a coordinate to a particular epoch, once you are there, you still have to remember the threshold of accuracy of all of the transformations and the original coordinates themselves. The transformation of data can never make it more accurate; in some cases, to get data from an older datum, two transformations of the data must be performed before HTDP time corrections can be applied. So, despite being very precise throughout the process, it is not always possible to be accurate.

Accounting for HTDP coordinates in the database of the software poses some design issues, as well. Should the definitions of time-based coordinates be categorized as coordinate transformations, or might they better be handled as coordinate systems? If they are handled as coordinate systems, the definitions will exponentially multiply as the new dated systems are added. HTDP allows for epoch changes down to an individual day, so for each reference frame, there are thousands of possible days and, therefore, thousands of different versions of a coordinate system to store in the database. This problem exists in any coordinate transformation application or library that binds coordinate transformation to datum definitions, and it also defines how the majority of coordinate-transformation software has been written.

Newer software tools incorporate the functionality of the HTDP utility and can aid in converting points from one reference frame to another, updating point coordinates to a different epoch and converting points between reference frames along with a change of epoch. Some of the most-advanced software allows the HTDP converter object to be automatically inserted as the datum shift with the date picker controls enabled, thus allowing the user to specify the epochs of the source and destination points in their respective reference frames. Such tools can facilitate time-dependent positioning tasks.

Improved Precision

Despite some of the continued roadblocks of working with time-based coordinates in currently available GIS software, time-based corrections allow us to take advantage of the highly precise positioning capabilities of modern GPS technology when introduced properly to the GIS data life cycle. Differential corrections, continuously operating reference stations, and real-time kinematic positioning systems allow us to record coordinates at amazing precision levels and observe the motion of the Earth’s crust over time. Until recently, the movements of the plates could only be approximated through conventional coordinate-transformation techniques that are only useful for localized areas and are out of date almost as soon as they are published.

These new, highly precise techniques are expanding capabilities from precision agriculture, earthquake modeling and land subsidence in coastal areas to traditional cadastral surveying and many other areas where precision is key. With recent advances in time-dependent positioning, we are now able to carry our data forward into the future as never before, and the capabilities are only increasing.

For Further Reading

• Leick, Alfred (2003), GPS Satellite Surveying, 3rd edition, John Wiley & Sons Inc., New York.

• Snay, Richard A. (1999), “Using the HTDP Software to Transform Spatial Coordinates Across Time and Between Reference Frames,” Surveying and Land Information Systems, Volume 59, Number 1, pp. 15-25.

• “Surveying & Positioning Guidance note 4” (2002), International Association of Oil & Gas Producers.

Author’s note: The source code and a PC-executable version of the NGS HTDP utility is freely distributed and may be downloaded at: