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A New Paradigm for High Accuracy Orbit Determination at the Centimeter Level

David W. Allan, Allan's TIME, Inc.
Fountain Green, UT 84632-0066

Neil Ashby, University of Colorado
Boulder, Co 89309

Gus R. German, Allan's TIME, Inc.
Fountain Green, UT 84632-0206

Invited paper presented at
ION GPS 2001
Salt Lake City, Utah
11-14 September 2001

click here for PDF copy (733kb)

If you print out this page, be sure to select "include images" so that the equations and figures will show.

Paper Contents:
Abstract Introduction The Physics Needed and the Concept for this New Paradigm  Some Practical Considerations  Error Analysis  Clocks  Summary  Perspective  Conclusion  

Figures, Table:
Figure 1 -- Doppler  Figure 2 -- Example at GPS Orbit  Figure 3 -- GPS Orbit  Table1

 Appendix A: Navigation and Timing Accuracy at the 30 Centimeter and Sub-Nanosecond Level
 Appendix B: Review of a New Unified Field Theory with a New Model of Gravity ( PDF)


Traditionally, satellite positions are determined by earth monitoring stations and their ephemeredes are given with respect to earth coordinates. As centimeter accuracies are anticipated we realize that the dynamics of the earth's variations in spin rate and radius (solid earth tides) are several centimeters from day to day.  A satellite constellation having centimeter accuracy would provide a robust opportunity to view the earth for the first time from an independent, local-inertial, reference frame.

Given the recent developments in atomic clocks and other relevant technologies, a new paradigm is developed herein utilizing currently available technology.  This paradigm leads us to believe that real-time centimeter accuracies are potentially achievable. This new paradigm takes advantage of the fundamentals in orbital physics to provide the necessary orthogonality and independence needed to potentially arrive at the desired robustness and accuracy, as well as independence for an orbiting constellation.


This paper provides an introduction to the concepts needed to determine satellite orbits with accuracy on the order of a centimeter and gives an idea of the fundamental physics involved. The theory has broad applicability, and many detailed studies would be necessary for any given application. We have used GPS orbits as an example.

About five years ago the ideas contained herein were conceived, and a patent application was filed jointly with support from Allan's TIME, Inc, Hewlett Packard Company, and the University of Colorado. The patent proceedings were discontinued due to an unusual set of circumstances, and the authors have chosen at this point in time to share the ideas -- so that they may be more broadly used. The abstract and the claims for this patent application are listed in Appendix A as filed some five years ago.

In view of the accuracy desired from the solution, the effects of relativity must be carefully considered. In the simulations performed in conjunction with the patent application, the relativity equations were calculated at the centimeter level. We will briefly discuss other effects that could affect such a satellite constellation at this level of accuracy.

The Physics Needed and the Concept for this New Paradigm

First, we need the first-order Doppler equation, and we will write it in a very simple form: y = v/c, where "y" is the normalized, Doppler-shifted, frequency offset as received at a monitor station coming from the atomic frequency standard on board a particular satellite (SV), "v" is the relative velocity between the SV and the same monitor station clock at a well surveyed point on the earth or otherwise, and c is the speed of light. Of course, if all the relativity effects are calculated and accommodated correctly, "y" will be zero when "v" is zero as specified in the equation. These zero values will occur uniquely at the point of closest approach -- in principle, providing a fiducial marker for the orbit.

In Figure 1 we show a simple diagram of the effect on this equation when measuring a Doppler frequency for an observer that is in line with the frequency of transmission. When the transmitter goes through the observer's position, two notable things happen: 1) the Doppler shift goes through zero, and 2) the derivative of the Doppler frequency offset, dy/dt, is a delta function. These distinctive characteristics of the Doppler transition, allow it to provide a fiducial marker for that point in time.

We can carry this concept into space with a highly accurate atomic clock on board a space vehicle (SV). Again, the Doppler frequency will be zero at the point of closest approach, and this point becomes a fiducial marker for the SV orbit.

The delta function from our initial case turns into a Doppler frequency slope, dy/dt, which will be a function of the SV's orbital geometry. As shown in Figure 2 for a GPS like orbit, this slope is dy = 1 x 10-13 per dt = 175 microseconds. Five years ago, this level of accuracy was easy to achieve. Now one can do much better. Frequency stability of 1 x 10-14 is commonly available on the new GPS block 2R rubidium clocks in space.

Next, we may simplistically write Kepler's third law in the following way:


T^2 = [(4pi^2)/(GMe)]r^3


where T is the period of the orbit, G is the universal gravitational constant, Me is the mass of the earth, and r is the radius from the center of the earth to the center of the SV. Now, we recognize in this simplistic circular orbit that the motion of the spacecraft is orthogonal to the radius vector. Hence, in Kepler's third law a natural orthogonality is built into the physics.

We can now ask the question, using a GPS orbit for example, if we know the SV's circular position to 175 microseconds out of a period of about 12 hours, how well do we know the radius vector, r? By taking the derivative of the above equation, we obtain the following:


dr/dt = 2/3 dT/T


where we can take dT = 175 microseconds, T is one orbit period of a GPS SV (about 12 hours), r is the radius vector from the center of the earth to the SV, and we solve for dr = 7 cm. This value of dr is an uncertainty on the absolute distance from the center of the earth to the center of the SV, and is totally independent of atmospheric delay variations to the extent that they are symmetric about the point of closest approach. Hence, we see the tremendous leverage factor this equation provides in determining the radius vector. This is illustrated in Figure 3. Also, we see that the improvement scales as the accuracy of the frequency standard in the SV.  So if we had an SV frequency standard with an accuracy of 1 x 10-14, then dr would be reduced to 7 mm. The technology currently exists to place this high level of accuracy in space. There are numerous other effects that need to be accounted for to prevent degradation of this level accuracy, but in concept one sees the tremendous advantage of this approach.

Some Practical Considerations

Because of the four dimensional nature of this high accuracy problem, one also has to deal with the estimations of all six of the orbital parameters the semi-major axis, the eccentricity, the initial longitude, the angle of perigee, the inclination angle, and the angle of ascending nodes -- in addition to the relativistic considerations.  At least two monitoring stations are needed in order to avoid ambiguities in determining these parameters.

In principle, all of the computations could be done in the SV. This would avoid having to communicate data from several monitor stations on the ground, which is currently a large segment of the overhead for GPS operation. For practical reasons, one may chose to perform the computations on the ground and then pass them up to the SV, or to perform them independently in both locations for redundancy.

One of the biggest practical considerations is to ensure that Equation l is valid. From a current science perception, this equation will be valid as long as the relativistic calculations are performed correctly and if the SV has no other forces than the gravitational force acting upon it. This can be accomplished to good approximation by having a "drag free" sensor at the center of mass of the SV, and then having small trim jets to keep it in a "drag free" state -- as has been done in other tests and as is planned for the "Gravity Probe-B" space experiment soon to be launched.

Error Analysis

In Table 1 we have made a preliminary error analysis of the various contributing factors. We have specifically made an estimate of the theoretical limit. As can be seen, most of the errors are of the order of a centimeter. The largest one being the error in the speed up and slow down of the earth's spin -- causing errors of several centimeters over the course of a day. The variations in the movement of the earth is, of course, one of the several measurements that one would like to study from this independent-local SV inertial frame. By looking at correlations across the constellation, one could sort out what the earth was doing versus what the errors were in the SV's positions. The earth rotates, at the equator, a centimeter in 22 microseconds. This would be resolvable, in principle, with an SV clock having an accuracy of 1x 10-14.

If the drag-free error contribution is given by 1/2 a t2, and the acceleration error "a" is given by 5 x 10-12 g, then in order to keep this error at one centimeter, the value of "t" is about 20,000 seconds -- less than six hours. Hence, the acceleration detectors would have to be updated about four times per day to contribute no more than one centimeter to the SV ephemeris error.

In typical satellite tracking, the vertical error component is the most difficult one to minimize. In this new paradigm, the vertical error component is the easiest. Currently, the center of the earth is not known with an accuracy of one centimeter. This new paradigm would significantly assist in improving the accuracy of this measure.


The atomic clocks that are being used, or that are being considered for space applications, and that would be useful for this new paradigm approach, are cesium beam, rubidium gas-cel1, and hydrogen maser designs. We will briefly touch on what we consider to be the best approach for each of these three types, picking from the best of the technology that is currently available.

About a decade ago, Hewlett Packard (Len Cutler, Robin Giffard, and colleagues) developed a revolutionary commercial, cesium-beam, frequency standard and clock, the model HP 5071A. As a result of the outstanding performance of this clock, the primary timing centers throughout the world have acquired them, and this type of clock now makes up 85 % of the weight of International Atomic time (TAI) and of the official civil time- (UTC) for the world. It has extremely high accuracy and typically negligible frequency drift. This technology has the potential to improve the performance of space cesium clocks by about an order of magnitude. There is no fundamental limiting reason why this technology couldn't be utilized in space. This technology provides outstanding long-term stability of the clocks and is the fundamental reason why these clocks have such a high weight in international timing. This same technology is now beginning to make a very important contribution to avionics positioning -- augmenting GPS. However, the short-term stability gives some limitation; it is about 6 x 10-12 tau-1/2.  This implies that one would have to average the frequency for about four days to have an uncertainty of about 1 x l0-14. There are ways to improve the short-term stability, and those are being. pursued. Because of the tau-1/2 behavior of the clock noise, whatever factor of improvement is gained, the square of that factor will be the decrease in time necessary to integrate to a particular level. In the above example, if 6 x 10-12 tau-1/2 were divided by a factor of 3 to a level of 2 x 10-12 tau-1/2 , then it would only take 1/9th of four days (less than 11 hours -- one GPS orbit) to reach an integrated level of 1 x 10-14 Fortunately, for this kind of noise (white-noise FM), one does not need to continuously monitor the time of the clock. In fact, the optimum estimate of the frequency is given by (x(t + tau) - x(t)/tau, where x(t) is the time deviation of the clock at time t.  These time deviation measurements could be one orbit apart if the precision of measurement was not a limiting factor (tau = 12 hours).

The GPS Block 2R rubidium clocks have been engineered to an astounding level, given the limiting physics associated with rubidium gas-cell technology. Their short-term stability is about 2 x 10-12 tau-1/2, which is about an order of magnitude better than most of the SV cesium clocks. Additionally, the flicker floor is lower than that for cesium -- at about 1 x 10-14. However, as can be expected from the fundamental physics of the devices, the frequency drift of the rubidium clocks is significantly higher than that of the cesium clocks. Despite this fact, the frequency drift of the rubidium clocks has been made remarkably low with very careful engineering. From the fundamental physics of the c1ocks, one would expect the long-term stability of the cesium-beam clocks to almost always be better than that of rubidium gas-cell clocks. The next generation rubidium clocks planned for the GPS III program have even better short-term stability -- about 7 x 10-13 tau-1/2, which means that in a single pass (tau = 10,000 s) one could integrate the clock noise down to a level of 1 x 10-14. This would be nearly ideal for this new paradigm concept for providing centimeter level accuracies for the SV orbits. However, the frequency drift would have to be very carefully calibrated with respect to the monitor station reference frequency standard, and each of the monitor station frequency standards would need an accuracy of 1 x 10-14.

Only one hydrogen-maser clock has flown in space.  This was the special clock built by Robert F. C. Vessot and colleagues for one of the finest tests of special and general relativity that has ever been conducted. The Swiss are proposing to provide a passive hydrogen-maser clock to be space qualified for the Galileo program. It will be very interesting to see how this program turns out. The short-term stability of active hydrogen masers can be made exceptionally good at about 1 x 10-13 tau-1This far out-performs any other commercial clock in the short-term and has some significant benefits in some applications -- as in the current new paradigm.  However, this is not what is being proposed, and wisely so, because making a space-qualified, active, hydrogen-maser c1ock with very reliable long-life performance is no small task. The Swiss proposal is much more reasonable, but has the disadvantage that the short-term stability is not nearly as good as for the active maser. It is nominally comparable to the current GPS block 2R rubidium clocks.


Below we list some of the advantages and needs of this new paradigm.


Real time accuracies approaching a centimeter.
Potential for autonomy; calculations could be performed onboard the SV.
Requires only a minimum of two tracking stations.
Has natural, heretofore under-utilized orthogonality from fundamental physics.
Is based on proven technologies.
With some of the new anticipated developments, this new paradigm has some very interesting application opportunities (see Appendix B).


Utilize the current high accuracy atomic clock technology
Utilize current "drag-free" technology


There are numerous applications where centimeter accuracy is desired, but the situation is limited by the accuracies afforded by current system architectures. GPS, as delivered now in real time, without other corrections, is pushing hard to approach one meter of accuracy. By differencing and double differencing the data with respect to known fixed points, which requires post-processing the data; then centimeter level accuracies are possible. By making use of this new paradigm and other new technologies now being developed, real-time accuracies at the centimeter level -- without other correction factors being needed -- will be possible.

Frequency can be measured more accurately than any other quantity. The length of the "second" (defined by a unique frequency in the cesium atom) is the most accurate measurement known to man. This new paradigm takes advantage of this fact. This new paradigm mitigates to a significant degree the large problem now facing most satellite tracking systems -- determining the absolute delay through the atmosphere. This delay is highly variable and unpredictable and is carrier-frequency dependent. If the uncertainty in this delay could be circumvented to some degree, this, by itself, would be a significant advantage. This new paradigm offers this circumvention to a significant degree.

As an example of current high accuracy measurements, the last six GPS block 2R rubidium clocks have long-term stabilities of about 1 x 10-14. At this outstanding level of performance, they are all experiencing common instabilities of about 2 x 10-14, with periods of the frequency instability of the order of three weeks, and with what appears to be close to a second harmonic is evident as well. However, it isn't a second harmonic because it isn't tied to the fundamental period. These observations appear to be common across the six most recently launched satellites. The searching question is what could be causing this phenomena? As we approach real-time centimeter accuracies, these kinds of questions will undoubtedly arise, and they may have very revealing answers.


This paper was given as a last-minute invited, fill-in paper to help take the place of those who could not come to ION GPS 2001 because of the horrific attack on the United States on 11 September by terrorists. Hence, there has not been time to do a full development of the material herein. It is our desire to give enough information so that the interested investigator can pursue this new paradigm efficiently and well.

When we performed the investigations for the patent application, relevant technology had not progressed as far as is now available. The current state of technology makes this new paradigm even more promising. It is believed by the authors that pursuing the improved levels of accuracy promised by this type of approach will greatly benefit the user community and also add significantly to scientific understanding of questions which were previously unanswerable.

Figure 1 -- Doppler

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Figure 2 -- Example at GPS Orbit

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Figure 3 -- GPS Orbit

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Table 1




 4 cm/12 hrs.


~ cm


~ cm


~ cm


 < cm


 < cm


t2 x 2.5 x 10-11 m/s2


 7 x l0-16


 4 x 10-17


~ cm


~ cm


~ cm


 < cm


 1 to 2 cm


~ cm


Involves trade-offs between several factors: clock noise, integration time, Kepler's 3rd law leverage factor, geometry of satellites with monitor stations, method of measuring ionosphere and troposphere and other factors

~ cm

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 Appendix A: Navigation and Timing Accuracy at the 30 Centimeter and Sub-Nanosecond Level

 Appendix B: Review of a New Unified Field Theory with a New Model of Gravity (PDF)


See also

click here for PDF copy of entire paper (733kb)
Synchronistic Modulation Detection - Incorporating new unified field theory for inexpensive, highly accurate navigation.  'Directions 2001' article by DWA published in GPS World, Dec. 2000.
List of other publications by David W. Allan




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