Experimental validation of a dual uplink multifrequency dispersive noise calibration scheme for Deep Space tracking

Authors


Corresponding author: G. Mariotti, Università di Bologna, Via Fontanelle 40, Forlì, FC 47121, Italy. (gilles.mariotti@unibo.it)

Abstract

[1] We discuss the implementation and effectiveness of a dispersive noise multifrequency calibration scheme for Deep Space tracking. We show that the combination of two phase-coherent links at X band and Ka-band, with two separate uplink carriers, can provide an effective plasma and ionospheric noise removal, in the order of 75% of the plasma noise affecting the Ka-band link. This algorithm, which we refer to as “Dual Uplink, Dual Downlink”, shows a modest loss in the radio link stability, if compared to the complete, state-of-the-art calibration achieved by a more complex radio system, which supplements the two separate uplinks and downlinks at X band and Ka-band with an additional “cross-link” (X-up/Ka-down). The calibration accuracy of these two algorithms is thoroughly compared to define their advantages and shortcomings. Finally, Cassini's multifrequency tracking data acquired in 2002 during a General Relativity solar conjunction experiment aimed at the estimation of the parametrized post Newtonian parameter γ were reanalyzed to assess the capability of the Dual Uplink, Dual Downlink calibration algorithm to support accurate radio science experiments.

1 Introduction

[2] Interplanetary plasma is a major noise source in the error budget of deep space tracking experiments. The reduction of its effect plays a key role when dealing with the scientific requirements of challenging experiments such as gravitational waves detection or relativity tests. The concept of multifrequency tracking of interplanetary spacecraft (S/C) for the removal of interplanetary plasma and ionospheric effects from Doppler observables was first presented by Bertotti et al. [1993] and applied to Ulysses tracking data [Bertotti et al., 1995]. Nowadays, multifrequency calibrations are also widely used in Global Navigation Satellite Systems applications for removing the ionospheric range delay affecting pseudorange and phase observables [Petit and Luzum, 2010]. In the next future, the BepiColombo Mercury Orbiter Radioscience Experiment will heavily rely on multifrequency radio tracking in order to achieve its ambitious error budget goals [Iess and Boscagli, 2001; Iess et al. 2009].

[3] The full multifrequency link configuration proposed in Bertotti et al. [1993] was successfully tested for the first time using Cassini's tracking data acquired during the cruise phase radio science experiments [Tortora et al., 2002, 2003, 2004; Bertotti et al., 2003; Armstrong et al., 2003], showing a dramatic improvement in data stability, if compared to the raw X band and Ka-band observables for data acquired at low solar elongations angles (or SEP, Sun-Earth-Probe angle). The residuals obtained by the calibrated observables using a simple orbital model (the six parameter fit, described by Bertotti et al. [1995] based upon the Doppler formulation introduced by Curkendall and McReynolds [1969]) showed a fractional frequency stability (the fluctuation of the received sky frequency around the carrier reference frequency, in terms of Allan standard deviation, ASDEV) compatible with the one achieved during solar oppositions, a benign scenario for interplanetary plasma noise.

[4] Using the formalism presented by Bertotti et al. [1993], we see that a phase-coherent carrier sent in a round-trip loop from Earth and back is subjected to a phase scintillation caused by the refractivity index of the interplanetary medium, that consists of fully ionized plasma and is crossed during the uplink and downlink legs. This refractive index can be described as inline image, assuming that the carrier frequency f is much higher than the electron plasma frequency: inline image, with e electron charge, ne electron number density, and me electron mass.

[5] The signal received by the ground station shows a total fractional frequency shift y and, under the assumption that the signal is affected only by dispersive noises, this is the combination of three components: the true observable ynd (the “non-dispersive” Doppler shift), the uplink dispersive contribution, and the downlink one:

display math(1)

where P and P are unknowns, related to the total electron content inline image of the medium and scaled with the uplink and downlink carrier frequencies, consistently with the electrostatic plasma dispersion relation.

[6] In a phase-coherent radio link, as the ones used for two-way tracking of deep space probes, the downlink reference frequency is not generated by the on-board oscillator, but keeps the ground station signal stability by simply scaling its frequency by a certain turn-around ratio α and transponding it back to Earth.

[7] Thus Equation (1) should be rewritten as

display math(2)

Since the goal of the calibration is to retrieve the ynd component from the received observable y, we have a total of three unknowns.

[8] The full multifrequency link calibration, as implemented for Cassini's radio science experiments, exploits a radio system capable of generating three different observables, with two phase-coherent links working at X band and Ka-band, respectively, and a third “cross-link” generated receiving the up-link at X band and scaling the carrier to Ka-band for the downlink leg. So for Cassini, we have three independent observables acquired at the ground station (X/X, Ka/Ka, and X/Ka), and as Equation (2) holds for each one, a 3 × 3 set of equations can be compiled:

display math(3)

Furthermore, Equation (3) can be simplified using inline image, inline image and inline image to get:

display math(4)

Equation (4) is easily inverted, and the three unknowns are solved for, to get in particular the non-dispersive (plasma-free) observable:

display math(5)

Despite its simplicity, this calibration scheme is extremely effective in terms of link stability and allows eliminating almost entirely its dependency upon the S/C SEP angle, as shown by Tortora et al. [ 2003, 2004]. The spectrum of solar plasma noise is strongly dependent on the SEP, because it represents the solar wind velocity component perpendicular to the carrier optical path, that is the main source of the phase scintillation [Armstrong et al., 1979; Asmar et al., 2005]. The price to be paid for these excellent results is the on-board hardware complexity: a radio system handling three different phase-coherent links, i.e., five different carriers spread between two bands, is quite demanding and expensive.

[9] A different, simpler calibration algorithm was still proposed by Bertotti et al. [1993] and its performance preliminarily assessed using qualitative considerations. The mathematical formulation of this simpler calibration scheme, and the comparison between the two algorithms are presented in the following section.

2 Dual Uplink Dual Downlink Calibration Scheme

2.1 Overview

[10] The plasma-free observable of Equation (5) is merely a linear combination of the received signals, and the coefficients are given by the standard values of turn-around ratios and reference frequencies in use at the Deep Space Network: αxx = 880 / 749, αxk = 3344 / 749, αkk = 14 / 15, and given a typical frequency ratio between X and Ka uplink frequencies β ≃ 320 / 75, we have that the main contribution to the plasma-free observable comes from the Ka/Ka link (as expected since higher frequencies are less affected by dispersive noises), while the cross-link component has a marginal role in the computation:

display math

One could speculate as to whether removing the cross-link from Equation (4) could simplify the on-board RF equipment, without reducing the calibration performance.

[11] If we delete the second equation from (4), we end up with an underdetermined set, in which the three unknowns cannot be solved for. However, it is possible to generate a new combination y* that is as close as possible to the real non-dispersive observable ynd. Bertotti et al. [1993] stated that with a convenient linear combination, this new observable removes completely one of the plasma contributions, while being slightly affected by the other. One can actually choose which of the two plasma amounts to cancel out completely by defining the y* combination accordingly: the most effective combination is the one that eliminates the uplink contribution.

display math(6)

However, the arbitrary removal of one of the two plasma components does not ensure by itself that the resulting y* has the lowest dispersive noise level achievable by an X/X+Ka/Ka combination: on the contrary, the coefficients of the linear combination should be computed through a statistical optimization of the variance of y*.

[12] This optimization, shown in detail in Appendix A, defines the Dual Uplink Dual Downlink combination:

display math(7)

with inline image.

[13] This combination retains fractions of both the uplink and downlink plasma contributions, but the overall dispersive effect is lower than the one affecting the combination shown in Equation (6).

2.2 Experimental Results

[14] One can note that every observable, both raw and calibrated, can be expressed as a sum of ynd, y, and y. Table 1 shows the numerical values of the coefficients used in Equations (4)-(7): these coefficients show how much an observable is affected by the three signals.

Table 1. Coefficients of Plasma Contents and the “Non-dispersive” Contribution for Each Observable
 yndyy
yxx110.7244
yxk110.0502
ykk10.04370.0502
ynd (Equation (5))100
y* (Equation (6))100.0194
y* (Equation (7))1 − 0.00910.0129

[15] It is clear that the residual plasma contribution to y* is very modest, and it can be concluded that the Dual Uplink Dual Downlink calibration brings down the total plasma noise standard deviation by about 75% when compared to the one affecting the raw Ka/Ka observable (assuming y and y having the same noise level: inline image.)

[16] It has to be noted that the value of β varies through different tracking passes, depending on the actual uplink frequencies needed to correctly track the probe. All numerical coefficients shown here assume β = 4.7828, the average value used for Cassini's solar conjunction experiment (SCE). A quick and coarse verification of this theoretical result is given in the following. Assuming plasma noise and plasma-free residuals are uncorrelated signals, the variance property (also valid for the Allan variance) holds:

display math

With KaPN and PN* indicating the plasma noise affecting the post-fit residuals generated by fitting the Ka/Ka observable and the one calibrated by the Dual Uplink link, respectively. These residuals are indicated by reskk and res*, along with the residuals of the fully calibrated observable, resnd.

[17] Exploiting these identities, we can write:

display math

and then compare this value to the expected one.

[18] This evaluation was carried out by analyzing Cassini's multifrequency tracking data acquired in the summer of 2002 during the first planned Solar Conjunction Experiment: the actual noise level ratio is shown in Figure 1.

Figure 1.

Ratio between the plasma noise levels affecting the Dual Uplink Dual Downlink and the raw Ka/Ka residuals.

[19] A much more interesting result comes from the comparison of the stability of the residuals, that highlights the actual gap in calibration performance between the full Multifrequency link and the Dual Uplink Dual Downlink scheme. The plot of the Allan standard deviation at 1000 s integration time (Figure 2) of y* obtained using Equation (7) and ynd obtained using Equation (5) shows an excellent level of phase stability for the simpler plasma calibration scheme: the Allan standard deviation is consistently below 3 · 10 − 14 s/s, and the degradation in stability during the days closer to the solar conjunction (that occurred on DOY 172/2002), is smaller than the Ka/Ka one.

Figure 2.

Allan standard deviation at 1000 s for the raw X/X and Ka/Ka links and the calibrated Cassini's SCE Doppler residuals. Doppler noise due to Earth troposphere, present in the original observables, was removed prior to this analysis in order to account only for dispersive noise sources. The gap in the data between DOY 168 and 173 is due to a malfunctioning of the Ka-band uplink transmitter.

[20] Moving towards larger SEP angles, where the plasma noise effect weakens, the difference between the calibration performance of the two algorithms (that ranges between 20% and 80%) should narrow: this allows relaxing the requirements on the design of the on-board RF chain, at least for missions that do not envisage critical operations at low SEP angles.

[21] For a further analysis of the residuals, the Allan standard deviation has been plotted as a function of different integration times τ [Barnes et al. 1971] in Figure 3 for DOY 161/2002 and in Figure 4 for DOY 168/2002. The slope of the loglog ASDEV plot, as well as the power spectrum plots, shows the nature of the various signals.

Figure 3.

Allan standard deviation of Doppler residuals versus integration time for DOY 161/2002. Integration time in seconds.

Figure 4.

Allan standard deviation of Doppler residuals versus integration time for DOY 168/2002. Integration time in seconds.

[22] Both figures show that the Multifrequency link calibrated residual follows a slope of about τ − 1 / 2, that corresponds to a white noise process in a fractional frequency shift time series (Barnes et al. [1971] reports the exact conversion between power spectral density (PSD) and Allan variance for several types of noises), while the raw Ka/Ka residuals ASDEV retains a τ − 1 / 6 slope, associated to a Kolmogorov spectrum in the phase signal (Armstrong et al. [1979] showed that solar wind and corona bear inside a fully developed turbulence, with a phase frequency spectrum Sϕ ∝ f − 8 / 3 that translates into an Allan variance of σ2 ∝ τ − 1 / 3, hence a τ − 1 / 6 slope in the Allan standard deviation). The Dual Uplink Dual Downlink shows a τ − 1 / 2 slope for DOY 161, the same as the Multifrequency link, indicating an almost complete removal of the dispersive noise, while for a day closer to conjunction, DOY 168, the slope of the Dual Uplink Dual Downlink is τ − 1 / 3, so we can infer that the calibration is not as effective as the full multifrequency link for this tracking pass.

[23] Finally, we focused our attention on the DOY 173 residuals: when fed with this data set, the Dual Uplink Dual Downlink calibration routine performed better than the full Multifrequency link one, which is supposed to provide a complete removal of dispersive noises. As a matter of fact, when the RF beam lies closer than 2–4 solar radii from the Sun, several additional effects, due to the solar corona, upon the RF signal (magnetic correction to the refractive index, dispersive beam displacement and diffraction) become non-negligible, and the model (1) becomes inadequate to describe the observables. Thanks to its formulation, the Dual Uplink Dual Downlink should be less sensitive to this modeling errors. However, the expected magnitude of these effects [Bertotti and Giampieri, 1998] is too small to be the cause of this behavior, since the impact parameter during DOY 173 was around 4 solar radii.

[24] A more plausible explanation of the unexpected result on DOY 173 could reside in the omission of X/Ka link: Cassini's on-board transponder, due to severe amplitude scintillation at X band, loses lock on the uplink signal and its phasor cannot be precisely reconstructed. The resulting fluctuations in the instantaneous phase are then multiplied by the αxk ratio (roughly 4.5 times) resulting in high noise levels for the X/Ka link.

3 Estimation of the Post Newtonian Parameter γ

[25] Cassini's multifrequency Doppler observables acquired during the SCE in 2002 allowed to obtain a significant reduction of the uncertainty of the parametrized post Newtonian (PPN) relativity parameter γ, describing how the space-time curvature induced by a gravitational mass is responsible of deflections and delays experienced by an RF wave. The solution γ = 1 + (2.1 ± 2.3) · 10 − 5 obtained by Bertotti et al. [2003] reduced the formal uncertainty with respect to previous experiments, thanks to the Multifrequency link and water vapor radiometer-based Earth troposphere calibrations. The very same analysis approach is replicated here with the only difference of using Doppler data that were calibrated using the Dual Uplink Dual Downlink algorithm, both to further quantify the loss of accuracy of this calibration scheme and to assess its impact for scientific applications, in addition to the analysis of the post-fit residuals discussed in the previous section. To this aim, the Orbit Determination Program (ODP) [Panagiotacoupulos et al. 1974; Moyer, 2003] of the Jet Propulsion Laboratory, Pasadena, California (JPL) was used to generate Cassini's orbital solution and to estimate the parameters of interest using a set-up identical to the one thoroughly described by Bertotti et al. [2003], that is using the same estimated and consider parameters to keep both results as homogeneous and comparable as possible. Our new results show that the estimation of γ is severely biased away from the unity value predicted by general relativity, γ = 1 + ( − 12.5 ± 2.6) · 10 − 5. While the formal uncertainty is very close to the one achieved using the plasma-free observables, the estimated value of 0.999875 is about 5 standard deviations off from the nominal solution, showing a bias due to the plasma noise that the Dual Uplink Dual Downlink calibration was unable to absorb. Figure 5 shows the average of the Dual Uplink Dual Downlink residuals for each tracking pass obtained by using the Bertotti et al. [2003] orbital solution as a priori value for a passthrough analysis (meaning that no iteration on the estimated parameters was carried out). The mean of residuals moves away from zero when approaching superior conjunction, in a very similar manner to the raw X/X uncalibrated observables, shown for comparison in Figure 6.

Figure 5.

ODP Doppler residuals of the Dual Uplink Dual Downlink calibrated observables.

Figure 6.

ODP Doppler residuals of the raw X/X observables.

[26] Note that this behavior of the Doppler residuals shows up only when no estimation iterations are performed, while a full orbit determination run (with estimation iterations) returns good zero-mean Doppler residuals, showing that the bias is transferred from the residuals to the estimated parameters, γ in particular.

[27] The estimation of γ is severely affected by solar plasma because both physical effects produce a deflection of the ray path, so that the residual dispersive noise left by the Dual Uplink Dual Downlink calibration is recognized by the filter as a gravitational bending. As these deflections have opposite sign [Bertotti and Giampieri, 1998], the resulting estimated value of γ is lower than 1 or, in other words, the bias is negative. Moreover, the bias should grow when using lower carrier frequencies.

[28] This statement is confirmed by a model for the bias generated combining the Baumbach-Allen model for the electron density ne of the solar corona [Bertotti et al. 1993] with the formulation of the gravitational delay from Moyer [2003]:

display math(8)

where μ is the gravitational parameter of the Sun, r1 and r2 the position vectors of the Earth and the S/C, and r12 the vector of the ray path. The coronal model has been used to estimate the path delay due to plasma δlpl, and then Equation (8) has been inverted to find γ using inline image on the left-hand side. This gives us a qualitative trend that we expect the ODP results to follow and, actually, the filter behaves accordingly: the estimates generated using these observables are γ = 1 + ( − 8.81 ± 0.94) · 10 − 3 for the raw X/X link and γ = 1 + ( − 3.23 ± 0.2) · 10 − 3 for the raw Ka/Ka link (Figure 7).

Figure 7.

Expected (black) and observed (red) γ bias.

[29] Although the results obtained using the Dual Uplink Dual Downlink calibration scheme are not promising for highly demanding radio science experiments, an attempt was made to combine the best of each calibration in a mixed data set from Cassini's 2002 SCE. The combined observables were formed starting from the original Multifrequency link calibrated data and replacing in it the Dual Uplink Dual Downlink observables for DOY 173 only, because of the peculiar ASDEV levels reported in Section 2.2. The rationale for this choice is that DOY 173 is the most important tracking pass in the data set, from the γ estimation point of view, since this parameter becomes more observable approaching conjunction (as stated by Iess et al. [1999] the gravitational influence on y is inversely proportional to the impact parameter). With this mixed data set input, our best estimation result is γ = 1 + (3.48 ± 2.17) · 10 − 5, with a 5% reduction of the formal estimation error, with respect to the reference result in Bertotti et al. [2003].

4 Conclusions

[30] We have shown the application of an optimized Dual Uplink Dual Downlink calibration scheme to Cassini's SCE Doppler data, in order to evaluate the loss of accuracy that occurs when using a X/X+Ka/Ka radio system instead of a fully equipped X/X+X/Ka+Ka/Ka system. The results are intended to help the definition of the best trade-off between the loss of accuracy and the reduced complexity of the RF system. The analysis showed that, in superior conjunction conditions, the corruption of radiometric observables due to interplanetary plasma is not entirely canceled by this algorithm, unlike the full multifrequency link calibration, that performs better than the Dual Uplink Dual Downlink method by a factor between 20% and 80% (in terms of Allan standard deviation of the orbital fit residuals) depending on the SEP angle. An exception is represented by the tracking pass of DOY 173 that was the closest to the superior conjunction: using this data, the Dual Uplink Dual Downlink performs slightly better than the triple link.

[31] Another test was carried out using a previous set-up for a relativity experiment aimed at the estimation of the parametrized post Newtonian (PPN) parameter γ. The calibration appears not to be suitable for such a challenging scenario since γ's estimation performed using observables calibrated by the Dual Uplink Dual Downlink algorithm shows a significant bias away from the unity value predicted by Einstein's theory.

Appendix A: Statistical Optimization of Dual Uplink Dual Downlink Calibration Algorithm

[32] Given the observable model from (4), a linear combination of X/X and Ka/Ka observables with coefficients χ and ψ:

display math

is also a linear combination of ynd, y and y:

display math

As we can neglect the cross-correlation among the three contributions (in particular the two plasma components are correlated only at a certain delay [Asmar et al. 2005]), the variance associated with this observable is

display math

[33] This identity can be further simplified by imposing that ynd should be carried without modifications into y* (thus requiring χ + ψ = 1), and using the hypothesis that the two plasma contributions have the same noise level: σ ≃ σ = σpl.

[34] With these assumptions, the previous equation becomes a function of χ only:

display math

[35] The optimization of the linear combination of the X/X and Ka/Ka observables consists in the minimization of the coefficient of inline image through an appropriate value of χ, computed from the minimum condition inline image.

[36] The final values of χ and ψ are then obtained:

display math

With inline image.

[37] Using Cassini's values, we have χ ≃ 1.0552 and ψ ≃ − 0.0552, while the formulation presented in Bertotti et al. [1993] uses χ ≃ 1.0457 and ψ ≃ − 0.0457: Figure A1 shows the comparison between the coefficients of inline image for the two algorithms.

Figure A1.

Coefficients of the uplink (red line) and downlink (blue line) plasma variance as a function of χ. The optimal Dual Uplink Dual Downlink (purple dot) minimizes the sum of the two plasma contributions to the variance. The formulation proposed in 1993 (red dot) cancels entirely the uplink signal but the total coefficient (black curve) is not at a minimum.

[38] To test the effectiveness of this theoretical statement, both dual uplink calibration schemes were applied to Cassini's SCE data. The optimum link has achieved a 10% average reduction of the ASDEV of the Doppler residuals at 1000 s integration time, with a peak of 30% for a few passes. This result is shown in Figure A2.

Figure A2.

Comparison of the Allan standard deviation at 1000 s integration time of the Doppler residuals for the 1993 (red) and optimized (black) Dual Uplink Dual Downlink calibration schemes. Plasma free residuals (magenta) obtained using the full Multifrequency calibration are shown for reference.

Acknowledgment

[39] This work was funded in part by the Italian Space Agency (ASI) through contract I/080/09/0.

Ancillary