We present an innovative technique that eliminates the phase noise of the onboard frequency reference as the main limiting factor on uplink radio occultations, while allowing full utilization of the expected improvement in SNR over downlink experiments. This technique relies on conducting simultaneous uplink and downlink occultation measurements to synthesize a phase reference of quality determined by the stability of the atomic clock at the ground station, eliminating the frequency fluctuations of the onboard clock as the limiting factor. Atomic frequency standards typically have frequency stability that is about 2–3 orders of magnitude better than space-qualified state-of-the-art clocks. The quantum leap improvement in reference phase stability and, simultaneously, in measurement SNR opens the door to fundamentally new science capabilities for radio occultation observations of planetary rings and atmospheres. In particular, the improvement is expected to enhance sensitivity to tenuous target regions, to increase the dynamic range within opaque regions, and to improve the spatial resolution of mapping large- and small-scale target structures.
 Spacecraft interplanetary tracking is the most sensitive technique to date for measuring distances and velocities of objects in the solar system, leading to information on masses and higher order moments of gravity fields of planets, their satellites, and asteroids. By analyzing the perturbations in phase and amplitude of a radio signal transmitted by a spacecraft and received at a ground antenna, determination of the physical states and chemical compositions of atmospheres, and the structure of rings that surround celestial bodies [Marouf et al., 1986; Tyler, 1987; Gresh et al., 1989] have also been inferred.
 One-way downlink occultations as described above, are usually preferred to two-way (Earth-spacecraft-Earth) coherent measurements essentially for two reasons. First, it is difficult to maintain a two-way radio link with the spacecraft when signal strength and frequency exhibit dynamic variations, as can occur during a typical occultation. Second, the time delay in establishing a two-way radio link after the downlink signal has been absent would result in a serious loss of data at egression from the satellite occultations.
 The accuracy in the reconstruction of the profile of the media the radio signal propagates through depends on the magnitude of the signal-to-noise ratio of the microwave link, as well as its stability in amplitude and frequency. In order to increase the signal-to-noise ratio during occultation experiments, it was first proposed by Tyler  to perform occultation experiments by transmitting a microwave signal from the ground to the spacecraft. By taking advantage of the higher transmitting power available at the ground antennas, data with a signal-to-noise ratio significantly higher than that achievable during ordinary downlink measurements can be obtained. With the use of a Doppler readout added to the spacecraft payload, amplitude and Doppler measurements can be performed on board, and data can be digitized, recorded, and transmitted to the ground at a later time of the mission.
 One-way measurements, whether performed in the uplink or downlink mode, suffer, however, from the frequency fluctuations of the onboard oscillator. Space-qualified oscillators, which are called ultrastable oscillators (USO), typically display a frequency stability that is about 2–3 orders of magnitudes worse than that of ground-based clocks [Asmar, 1996; Vessot, 1996].
 In this paper we will show that it is possible to synthesize a new Doppler data set by linearly combining the two one-way Doppler measurements (performed on the ground and on board, respectively) in such a way to remove the frequency fluctuations of the USO. An outline of the paper is given here.
 In section 2, after deriving the expressions of the two one-way Doppler measurements recorded on the ground and on board the spacecraft, we point out that the frequency fluctuations due to the onboard and ground clocks enter into the two data sets with well-defined transfer functions. By properly time-shifting the Doppler data recorded on board with respect to the ground data and linearly combining them, we derive a new data set from which the frequency fluctuations of the USO are entirely removed. As a result of this operation, however, the frequency fluctuations due to the media the radio signal propagates through appear at two different times in this synthesized data, separated by twice the distance between the spacecraft and the occulted media. This effect can be deconvolved from the data and the information provided by the frequency fluctuations due to the occulted media can be recovered.
 Since the implementation of this technique requires knowledge of the separation between the Earth and the spacecraft, and synchronization between the Earth and the onboard clocks, in section 3 we estimate the required accuracies in the knowledge of these quantities. In section 4 we finally present our comments, and conclude that the technique proposed in this paper could be implemented successfully with presently existing technology.
2. One-Way Doppler Responses
 In what follows we will derive a method for removing the frequency fluctuations of the USO. This relies on augmenting the radio system onboard the spacecraft with Doppler recording and transmitting capability (see Figure 1). This hardware, usually referred to as the onboard Doppler readout, measures the difference between the phase of the received signal and the phase of the onboard signal referenced to the ultrastable oscillator. It then digitizes these measurements, and applies a time tag to each sample. The quality and format of the data recorded on board is essentially identical to those generated at the ground station during the experiment. Such instrumentation has already been used by the Galileo mission for performing onboard Doppler measurements of the microwave signal transmitted by the Galileo probe while it was descending into the Jupiter atmosphere. The cost involved with the addition of such hardware represents a small fraction of the overall cost of the onboard radio equipment.
 With this radio configuration, one-way Doppler can therefore be measured on board the spacecraft, recorded, and telemetered to the ground at a later time during the mission. By linearly combining the Doppler measurements performed on board with the one-way Doppler data measured at the same time on the ground, the frequency fluctuations of the onboard clock can be removed, as we will show below.
 Let us assume that the two clocks, at the ground station and on board the spacecraft, are perfectly synchronized, and let us also assume for sake of simplicity that the uplink and downlink signals have the same frequency ν0. If we denote with yE (t) and yS (t) the one-way Doppler measurements performed at the same time t at the ground station and on board, respectively, they are given by the following rather complete expressions [Tinto, 1996; Armstrong, 1998]:
where we have denoted with CE (t) the random process associated with the relative frequency fluctuations of the clock on the Earth, CS (t) the frequency fluctuations of the USO, B (t) the joint effect of the noise from buffeting of the probe by non gravitational forces and from the antenna of the spacecraft, T (t) the joint frequency fluctuations due to the troposphere, ionosphere and ground antenna, AE (t) the noise of the radio transmitter on the ground, AS (t) the noise of the onboard radio transmitter, ELE (t), ELS (t) the noises from the electronics at the ground station and on the spacecraft, and PE (t), PS (t) the frequency fluctuations due to the interplanetary plasma [Tinto, 1996; Armstrong, 1998]. In equations (1) and (2)L is the distance Earth-spacecraft measured in seconds (the speed of light c has been taken to be equal to 1), η(t) represents the frequency fluctuations introduced by the media the radio beams propagate through, and x is the distance between the spacecraft and the occulted media (see Figure 2).
 The frequency fluctuations due to the onboard clock can systematically be removed by generating the following linear combination of the two one-way Doppler measurements:
Equation (4) shows that the frequency fluctuations due to the USO have been entirely removed, and furthermore that the new observable Σ(t), obtained by delaying and combining the two one-way measurements, is equivalent to a coherent two-way measurement [Tinto, 1996].
 If we now denote with the Fourier transform of the data Σ(t), which we define as
from equation (4) we derive the following expression for the Fourier components of Σ(t):
From equation (6) we note that the Fourier components of the field η enter into the Doppler observable Σ through the transfer function cos(2πfx). For frequencies f such that fx ≪ 1, the modulation of can be neglected. For arbitrary frequencies, however (different from the frequencies where the transfer function cos[2πfx] has nulls), we can in principle remove the cosinusoidal by dividing the Fourier components of Σ(t) cos[2πfx] at each frequency f. In other words, except for a finite number of selected frequencies fk=(2k + 1)/4x, k = 1,2.., in the accessible frequency band, the frequency fluctuations introduced by the occulted media can be measured at a higher sensitivity level than that provided by each one-way measurement alone. This is because the frequency fluctuations of the USO have been removed by using the new data se modulation of Σ(t). Our technique enhances the quality of the uplink data in that it allows the experimenters to remove the frequency fluctuations of the onboard clock by combining the two one-way Doppler measurements (equation (3)).
 The expression derived above for assumes the relative distance x between the occulted media and the spacecraft to be constant. In general, this will not be the case. For a given trajectory, and a given frequency band over which the frequency fluctuations are measured, however, we can identify an integration time, τ, during which the separation between the spacecraft and the occulted media can be considered constant. In order to derive a relationship between the integration time τ and the corresponding variation of the distance x, which we call δx, let us consider the frequencies fk where the transfer function of goes to zero. If these frequencies will move by an amount larger than the frequency resolution (1/τ), we can argue that the transfer function of is no longer equal to cos(2πfx). Since |δfk| = fk δx/x, by imposing the condition of having |δfk| ≤ 1/τ, we find the following relationship:
If we assume, for instance, the spacecraft velocity v relative to the occulted planet to be constant over the timescale τ, equation (7) implies the following time τmax of integration:
As a numerical example, if we take v to be equal to 1.0 km/s, x = 3 × 105 km, and the frequency fk = 1 Hz, from equation (8) we find that the integration time τ should be less than about 550 s in order to apply successfully the signal's demodulation scheme explained earlier. The distance and velocity used in the numerical example above are representative of a possible trajectory followed by the upcoming Pluto-Kuiper Belt mission during its flyby at the planet Pluto. NASA has recently announced its intentions of funding a mission to Pluto, and for this reason we will concentrate on such a case in this paper. The choice of the frequency fk equal to 1.0 Hz is determined by the following consideration. The thermal noise at Ka-band (32 GHz) that affects the Doppler measurement Σ(t) starts to become larger than the USO noise itself at about this frequency, making unnecessary the implementation of our USO noise cancellation scheme. The estimate for the frequency given above assumes a distance of 33.5 AU, and a ground and onboard radio configuration similar to the one that will be used by the Cassini mission and augmented with an onboard Doppler readout.
3. Accuracy of the Method and Estimated Improvements
 The real limitations on the procedure described above, however, come from the remaining noise sources affecting the two phase measurements, the accuracy in the determination of the distance L, and the accuracy in the synchronization of the ground and onboard clocks. We will estimate below how these errors affect the tolerance of the method. In what follows we will assume the distances x, and L to be constant. Our analysis will identify the timescale during which this assumption is correct.
 The derivation of our method for removing the relative frequency fluctuations of the USO relied on the assumption of knowing exactly the Earth-spacecraft range L, and the clocks to be perfectly synchronized. If we assume instead the distance L to be known with an accuracy δL, and the clocks to by synchronized within a time accuracy δt, respectively, the cancellation of the frequency fluctuations of the USO in the data Σ(t) is no longer exact. In order to estimate the magnitude of the onboard clock fluctuations remaining in the data set Σ(t), let us define , , to be the estimated range, and the spacecraft time interpreted as the time t when the data yE(t) is recorded on the ground, respectively. They are related to the true range L and time t, by the following expressions:
If we now substitute equations (9) and (10) into equation (4), and expand it to first order in δL and δt, we obtain the following expression for Σ(t):
where S represents the time derivative of the random process CS. The technique described in this paper can therefore be considered effective if the magnitude of the remaining fluctuations from the USO are smaller than the fluctuations due to the other noise sources entering in Σ(t). This requirement implies an upper limit in the accuracies δt and δL. In order to estimate the magnitude of the required accuracies, let us focus our attention on the two terms entering into equation (11), associated with the frequency fluctuations CS (t) and the other noise sources
If we denote with , the Fourier transforms of the random processes ΔCS(t), N(t), respectively, from equations (12) and (13) we find that they are equal to
Equations (14) and (15) imply the following expressions for the one-sided power spectral densities of the noises ΔCS, N:
where the random processes CE, T, B, AS, AE, ELE, ELS, PE, and PS have been assumed to be uncorrelated, and we have introduced the notation ΓX for representing the one-sided power spectral density of the random process X.
 Since our noise cancellation algorithm can be considered effective if ΓΔCS (f) ≤ ΓN (f), from equations (16) and (17) we derive the following constraint on the accuracies δL, δt:
As an example application of equation (18), let us assume δL = −δt ≡ ρ. It is easy then to derive the following inequality for |ρ|:
In order to estimate the accuracy ρ, we will consider as an example a spacecraft with a radio instrumentation similar to that adopted for the mission Cassini, and that swill perform a radio occultation of the planet Pluto.
 The one-sided power spectral densities of the frequency fluctuations entering into Σ(t) at Ka-band (32 GHz) have been given by Tinto  and Armstrong  and are provided in Appendix A.
 After substituting into the right-hand side of equation (19) the expressions provided in Appendix A for ΓN, and ΓCS, it is easy to calculate numerically the minima of this function in terms of the frequency f, spacecraft distance L, and microwave frequency ν0. In the case of a mission to Pluto, for instance, the distance L is approximately equal to 33.5 AU, implying a minimum at about 0.6 s at the frequency f = 0.6 Hz when Ka-band is used. In other words, the distance to the spacecraft, and the synchronization of the two clocks, should be known within the above accuracies or better in order to effectively remove the frequency fluctuations of the USO from the combined data Σ(t). Since ranging accuracies at levels of a few hundred nanoseconds or better are routinely obtained with NASA's Deep Space Network when tracking interplanetary spacecraft [Ruggier et al., 2000], as a consequence it follows that clocks can also be synchronized within comparable accuracies, making the technique we have presented in this paper possible.
 In relation to the accuracies derived above, it is interesting to calculate the timescale during which the distance L will change by an amount equal to the accuracy itself. This identifies the minimum time required before updating the distance L(t) during the implementation of our technique. As an example, let us consider again an occultation taking place during a flyby of Pluto. Assuming a spacecraft velocity relative to the ground station equal to about 10 km/s, we conclude that the distance to the spacecraft L entering into the linear combination given in equation (3) will change by about 1.8 × 105 km over a timescale of about 5 hours. We conclude therefore that for the considered velocity of the spacecraft the distance L can essentially be treated as constant during such a timescale.
 We presented a time domain procedure for accurately cancelling frequency fluctuations introduced by an onboard frequency standard into Doppler measurements performed during interplanetary spacecraft radio occultations. The method involves separately measuring phase changes on the ground as well as on board. By suitable offsetting and differencing of these two time series, the common noise due to the USO is cancelled exactly (equation (4)).
 The effect of this procedure is to introduce a characteristic signature in the frequency fluctuations due to the media the radio beams propagate through. This effect can be deconvolved from the resulting data by proper signal-processing.
 To cancel the noise of the USO to the level of the secondary noise sources with this procedure, the Earth-spacecraft distance and the synchronization of the two clocks must be known with an accuracy of about 0.6 s. To demonstrate practicality of the method, we presented a general analysis of the required accuracies. We used estimates of the one-sided spectra of the frequency fluctuations of the noises expected to affect the Doppler sensitivity of a mission to Pluto, and estimated the required accuracies as functions of the Fourier frequency. We have found that the noise cancellation technique presented in this paper could be implemented with existing radio hardware technology.
 The improvement in reference phase stability and, simultaneously, in measurement SNR, should open the door to fundamentally new science capabilities for radio occultation observations of planetary rings and atmosphere. In particular, the improvement is expected to enhance sensitivity to tenuous target regions, to increase the dynamic range within opaque regions, and to improve the spatial resolution of mapping large- and small-scale target structures. A quantitative analysis of such improvements will be the topic of a forthcoming paper.
Appendix A.: Noise Sources
 In this appendix we provide a description of the radio hardware needed for implementing the technique described in the main body of this paper, the corresponding one-sided power spectral densities of the frequency fluctuations introduced by these subsystems into the observable Σ(t), and discuss the frequency fluctuations due to the Earth atmosphere, ionosphere, and the interplanetary plasma. For a more comprehensive analysis on the radio hardware the reader is referred to Tinto  and Tinto , while a review on the propagation noises is given by Armstrong .
 The master clock [Dick and Wang, 1999] and frequency distribution [Tinto, 1996, 2000] represent the overall contribution of the reference clock itself and the cabling system that takes the signal generated by the master clock to the antenna. This can be located several kilometers away from the site of the clock, implying that the need of a highly stable cabling system is required. It has been shown at JPL that optical fiber cables would not degrade significantly the frequency stability of the signal generated by the master clock. The corresponding one-sided power spectral density of the frequency fluctuations, introduced by these two noise sources, is equal to:
The first term on the right-hand side of the first line of the equation above provides the one-sided power spectral density of the frequency fluctuations due to the hydrogen maser, while the first term on the second line of the same equation gives the one-sided power spectral density of a compensated sapphire oscillator (CSO) [Dick and Wang, 1999] which is used as a cleanup loop for improving the frequency stability of the maser over the given Fourier band of interest. The remaining term, common to the two lines of the above equation, represents the one-sided power spectral density of the frequency fluctuations due to the frequency distribution.
 The onboard frequency standard provides the frequency and timing reference for the onboard radio instrumentation. Cassini, for instance, will rely on a crystal oscillator whose temperature is oven controlled in order to minimize the frequency fluctuations of the signal it generates. The one-sided power spectral density of the frequency fluctuations is given by the following expression [Asmar, 1996]:
 The transmitter chain includes all the frequency multipliers that are needed to generate the desired frequency of the transmitted radio signal, starting from the frequency provided by the clocks on board and on the ground. It also takes into account the radio amplifier, and the extra phase delay changes occurring between the amplifier and the feed cone of the antenna. The noise due to the amplifier is the dominant one, and its characterization is given by Tinto [1996, and references therein]. The one-sided power spectral densities of the frequency fluctuations are given by
The noises introduced by the receiver chains at the ground station and on board the spacecraft can be modeled as white phase fluctuations. The contribution to the overall noise budget from the receiver chain on the ground can be repartitioned into thermal noise (finiteness of the signal-to-noise ratio) and fluctuations introduced into the signal as it propagates through the cables and waveguides running from the feed of the antenna to the actual receiver. The effects of the latter noise source will be minimized with the use of future beam waveguide (BWG) antennas. These new antennas will be implemented by the year 2004 by the NASA Deep Space Network at its three sites: in North America (Goldstone, California), Europe (Madrid, Spain), and Australia (Canberra). One of these antennas is already operational at the California site, and it will be used during the Cassini radio science experiments. The frequency fluctuations of the receiver chain on board the spacecraft is estimated to be entirely due to thermal noise because of a simpler cabling system. The Doppler readout system on board the spacecraft (denoted with the circled times symbol in Figure 1) has already been built for the Galileo orbiter. A revised design of this Doppler readout system will be incorporated in the design of a new transponder that will be flown on several upcoming deep space missions (D. Antsos and S. Kayalar, Jet Propulsion Laboratory, internal report, 1999). Its function is to measure the difference between the phase of the received signal and the phase of the onboard signal referenced to the USO, digitize this observable, and apply a time tag to each sample before recording it. The format, quality, and bandwidth of the recorded data are essentially identical to those measured at the ground station.
 Under the assumption of using a 34 m diameter beam waveguide antenna that transmits with an 800 W Ka-band (32 GHz) amplifier to a spacecraft that is out to Pluto (at a distance of about 33.5 AU), a ground system noise temperature of about 70 K, an onboard Ka-band amplifier of 10 W, a system noise temperature on the spacecraft of 400 K, and a spacecraft high-gain antenna (HGA) with a diameter of about 4 m, we find the following one-sided power spectral density of the frequency fluctuations at Ka-band [Tinto, 1996]:
The buffeting of the spacecraft will introduce unwanted frequency fluctuations in the one-way Doppler observable. Estimates of its magnitude have been given by A. L. Riley et al. (Jet Propulsion Laboratory, internal report, 1990), and the one-sided power spectral density of the frequency fluctuations is given by the following expression:
The noise introduced into the Doppler observables by the Earth atmosphere and ionosphere, and by the scintillation of the interplanetary plasma, have been studied extensively in the literature [Armstrong et al., 1979; Armstrong, 1998; Keihm, 1995]. The scintillations introduced into the Doppler observables by the atmosphere are independent of the microwave frequency at which the spacecraft occultations take place. Since the technique of USO noise cancellation presented in this paper will be useful at Fourier frequencies smaller than about 1 Hz because of excessive thermal noise at higher frequencies, in the frequency band where our technique will be effective the one-sided power spectral density of the noise due to the atmosphere can be written as follows [Linfield, 1998]:
The first term in the equation above accounts for the remaining effect of the atmosphere after eighty percent calibration is applied to the data with the use of a water vapor radiometer [Armstrong, 1998], while the second term accounts for the effect of aperture averaging, which causes a reduction in delay fluctuations on timescales less than the antenna wind speed crossing time (1 to 10 s) [Linfield, 1998].
 The noise level introduced by the ionosphere and interplanetary plasma depends on the carrier frequency used during the tracking, as well as the Sun-Earth-Probe (SEP) elongation angle [Armstrong et al., 1979]. The one-sided power spectral density of the frequency fluctuations due to the plasma noise scales as the inverse fourth power of the carrier frequency, making the use of higher carrier frequencies a natural way for minimizing the effects of this noise source. Furthermore, since the one-sided power spectral density of the frequency fluctuations varies by almost seven orders of magnitude as the SEP angle changes from 1° to 175° [Armstrong et al., 1979], timing an occultation experiment in such a way that for a given carrier frequency and SEP angle the fluctuations introduced by the plasma are significantly smaller that the USO noise, should be a priority when designing a mission that will utilize the cancellation scheme proposed in this paper.
 The one-sided power spectral density of the frequency fluctuations given below assumes Ka-band carrier frequency, and an SEP angle of about 175°. Scaling to S (2.3 GHz) or X (8.4 GHz) bands, and to different SEP angles, can be performed as specified by Armstrong et al. .
 I would like to thank Essam A. Marouf for many stimulating conversations on spacecraft radio occultations. This research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.