## Doppler
Measurement Error
The Doppler tracking
of the craft is carried out in exactly the same way as described in Detection of the Pioneer
Anomaly in Doppler Data. The
frequency of a returned signal from a craft is compared to that of the frequency
of the sent signal. If the craft is
moving relative to the ground station then the frequency of the returned
signal will be different to that sent.
If the difference is not what is expected then some unknown
acceleration could be changing the velocity of the craft. This method has
associated errors which are inherent to the system and fundamentally limit
the accuracy of Doppler measurements.
Two of the main errors are presented below in detail. To summarise: if a craft is observed for
1000 seconds the two sources of errors studied in this section contribute to
an error of 0.0006 mm/s (X-band) on the Doppler measurement. This error comes about due to the choices
of the project design. Choices to do
with observation time, radio signal strength and noise levels. Which for an
unknown acceleration the size of the Pioneer Anomaly provides an observed
Anomaly Doppler velocity of 0.0009±0.0006 mm/s. Looking at this
and information presented below it is obvious that current technology is
capable of detecting the Pioneer Anomaly. The accuracy with
which the gradient of a Doppler velocity against time plot can be calculated
is dependent on the errors on the individual Doppler shift frequencies; each
data point. The error on the Doppler shift
propagates to the Doppler residual and then onto the Doppler velocity. The error on the
Doppler measurement is given by a standard deviation on the Doppler
velocity. This section of the project characterises
the magnitudes of some Doppler errors in X-band and then considers in the
light of these how the accuracy of interpreting the observed data is
constrained. The signal
received during an integration time is characteristic of an additive white
Gaussian noise source (AWGN). The AWGN
is due to the fact that the Doppler velocity comes from a counting process
and the result forms a Gaussian profile which has a statistically inherent
distribution; Doppler jitter. The
distribution of the Gaussian is characterised by the standard deviation of
the profile which can be calculated as a function of parameters associated
with the telemetry link; equation 1.
(1) The telemetry link
parameters designed by this project which are appropriate to determine
the Doppler jitter using equation 1 are shown in table 1.
Table 1 shows a
Doppler jitter of 0.011 mm/s. This can
be compared to the quoted value of 0.03 mm/s for the NASA Deep Space Network Along with
Doppler jitter there is an error on the Doppler velocity measurement due to
the instability of the generated reference frequencies. The accuracy to which the reference
frequency is generated is characterised by the Allan deviation of the
oscillator (σ
(2) The Allan
deviation is a function of integration time, and is given for systems with
respect to standard Doppler integration times. Currently Allan deviations vary from 10 The Allan
deviation adds an error to the Doppler velocity, σ
(3)vii Equation 3 gives
the error on the Doppler velocity over an observation time with number of
samples N. For the current design the
error due to Allan deviation at the ground station is 0.0001 mm/s during one
integration period of 1000s.
Doppler jitter
and Allan deviation are the two main inherent sources of noise
considered. Table 3 summarises the
values of the errors on the measured Doppler velocity over an integration time
of 1000s.
The error on a measured
Doppler velocity due to Allan deviation is nine times smaller than the
Pioneer Anomaly Doppler velocity and two thirds the size for Doppler
jitter. Therefore in one measurement
the Pioneer Anomaly can potentially be observed but not with a high level of
accuracy. To achieve higher accuracy
measurements are made over an observation time collecting a set of Doppler
velocities. Equation 1 shows
that Doppler jitter is independent of observation time and equation 3 shows
that the error due to Allan deviation is proportional to one over the square
root of the observation time.
Therefore during the observation time the error on the Doppler
velocity measurement decreases.
Graph 1 shows
that below 1000 seconds the inherent Doppler errors mask the Pioneer Anomaly
Doppler velocity but once the craft has been tracked for over 1000 seconds
(one integration period) the Doppler error values are below that due to the
Pioneer Anomaly. |

[1] *Radiometric
Tracking Techniques for Deep Space
Navigation*, JPL Deep Space Communications and Navigation Series, Wiley

[2] Nieto,
M.N., Turyshev, S.G., 2004, ‘Finding the Origin of the Pioneer Anomaly’, Class.
Quantum Grav. **21**, 4005-4023