i
Kalman Filtering:
Theory and Practice
Using MATLAB
Fourth Edition
Appendix C:
Feasibility Analysis of Doppler-based
Satellite Navigation
Mohinder S. Grewal
California State University at Fullerton
Angus P. Andrews
Retired from Rockwell Science Center, Thousand Oaks, California
Copyright 2015 by John Wiley & Sons, Inc.
ii
PREFACE
This appendix is intended to augment the examples used in [2] for demonstrating how one goes about assessing
the accuracy of timing-based satellite navigation, by providing an example for its now-obsolete predecessor: Doppler-
based satellite navigati on. It was first analyzed before the Kalman filter, and it provides an example that does not
need the Riccati equation for its performance analysis.
A simple model is used to demonstrate the feasibility of determining the longitude and latitude of a receiver
antenna from measurements of Doppler frequency shifts during a single overhead pass of a satellite with known orbit.
This was first demonstrated by William Guier and George Weiffenbach in 1 958, leading to development of the Transit
Navigation System, the world’s first satellite navigation system. This example also demonstrates how the expected
accuracy of Doppler-based satellite navigation depends on the relative location of the receiver antenna with respect
to the satellite ground track, and on Doppler measurement a ccuracy.
This sort of “prelim inary” analysis was ori ginally used for determining whether Doppler-based satell ite navigation
was feasible, from the standpoint of whether it could provide a navigati on solution for a fixed receiver location, and for
assessing how accurate that solution might be. As such, the approach does not require the full capabilities of a Kalman
filter for tracking a moving receiver with uncertain dynamics—as is now done for timing-based satellite naviga tion.
This may have been fortunate at the time (late 1950s), because the Kalman filter [6] had not been introduced yet.
However, these capabilities were soon added befo re Doppler-based satellite navigation became operational.
Appendix C
Feasibility Analysis of Doppler-based
Satellite Navigation
If you want to find the secret s of the universe, think in ter m s of energy, frequency and vibration.
— Nikola Tesla (1856–1943)
C.1 Historical Background
C.1.1 The Doppler Effect
This effect is na med for Austrian-born Christian Andreas Doppler (1803–1853), who developed a mathematical model
for the observed phenom enon that the frequencies of acoustic signals are shifted upward when the source is coming
toward the observer and downward when the source is going away from the observer. It is the same effect that—at
radar frequencies—i s used by traffic police to measure the speeds of automo biles on highways, and for Doppler-based
satellite navigation.
C.1.2 Satellite Navigation
Earth’s natural satellite—our moon—is thought to have been launched by an interplanetary collision around 4 billion
years ago.
Sputnik I, the world’s first artificial satellite, was launched by the former Soviet Union on October 4, 1957. This
happ ened on a Friday evening on the East Coast of the United States, when nearly everyone had gone home for the
weekend. When William Guier and George Weiffenbach showed up for work on the f ollowing Monday at the Applied
Physics Laboratory (APL) of Johns Hopkins University, the Soviet satellite was the main topic of discussion at lunch
in the cafeteria [5]. The two transmi ssion frequencies a round 20 MHz and 40 MHz had already been determined.
Weiffenbach was in possession of a good 20 MHz receiver, so he and Gui er decided to set up an antenna so they
could “listen in” after lunch. The lower satellite transmission frequency was found to be offset a bout 1 kHz from
20 MHz, and APL was near enough to the 20 MHz standard broadcast by the National Bureau o f Standards from
station WWV that they could use it to remove the 20 MHz component to better observe the Doppler variati ons
as the satellite passed from horizon to horizon. Weiffenbach had a spectrum anal yzer he had acquired for hi s PhD
thesis on microwave spectroscopy, so he was able to measure and record the Doppler variations from the nomina l 1
kHz offset frequency relatively accurately. As the satellite passed overhead from horizon to horizon, it was observed
to vary in frequency from about 50 0 Hz to 1500 Hz due to D oppler shift.
At that point, the selection of Doppler measurements had been determined by equipment availability, but it
would turn out to be surprisingly fortuitous. Guier was curious to determine whether the satellite orbit could be
determined from the Doppler variations alone, and was able to show that it could if the location of the receiver
antenna was known (which it was).
Although the batteries aboard the Sputnik I satellite lasted only 23 days in orbit, Guier and Weiffenbach were
able to continue their investigations for several months using the data recorded while it was transmitting. They also
had the use of the Univa c 1103A computer recently acquired by APL in their studies. In the process, they discovered
1
2 APPENDIX C. FEASIBILITY ANALYSIS OF DOPPLER-BASED SATELLITE NAVIGATION
that the influence of low-altitude gravity anom alies on the Sputnik I trajectories could not be ignored, but that the
same Doppler measurements could al so be used to estimate the gravity anomalies a nd provide a relatively accurate
ephemeris of the short-term satellite trajectory.
When these results were reviewed by Frank McClure at APL in March of 1958, he asked whether the problem
could be inverted. That is, if the orbit were known, could the receiver location be determined from the Doppler
measurements alone?
The answer turned out to be in the affirmative, and it was this discovery that led to the development of the first
satellite navigation system [ 5].
A need for such a navigation system had been established a few years earlier, when the U. S. Navy commissioned
a fleet of nuclear-powered, nuclear-missile-carrying submarines as a strategic deterrent in the Cold War against the
USSR and its allies. However, the best inertial naviga tion technology at that time was not sufficiently accurate for
launching ballistic missiles a fter many days at sea. Fortunately, the predicted accuracy of Doppler-based satellite
navigation would solve that problem. Furthermore, the antenna exp osure required fo r getting a posi tion fix while
submerged was deemed an acceptable risk if only Doppler frequency shift was to be used. Also, because the antenna
would be essentially a t sea level, only longitude and latitude had to be determined.
The Navy had studied the use of satellites as early as 1945 [1], but had fo und no justifiable application. Based
on the findings at APL, a Navy project to develop a satell ite navigation system based on Doppler measurements
was initiated in 1958, and would achieve op erational status in the 1960s [4]. The operational system would be called
Transit, or NAVSAT (for Navigational Satellite) [7].
C.2 Methods f or Proving Determinancy
Frank McClure’s question to Guier and Weiffenbach was whether the location of a receiver antenna at sea level could
be determined from the Doppler frequency variations in the signal of a satellite passing within view. The following
is an account of a progression of mathematical methods for resolving such issues.
These derivations use a si mple spherical-Earth model for the purpose of assessing the expected accuracy of
Doppler-based satell ite navigation, and for determining how the resulting navigational accuracy depends on the
receiver antenna location relative to the satellite ground track, and on various noise sources. This model would not
be good enough for the full Kalman filter implementation of the navigation solution, but it is adequate for statistical
analysis of the resulting navigation errors o f a more faithful Kalm an filter model [6].
C.2.1 Least Squares Approach
The feasibility of satellite navigation using Doppler measurements was first demonstrated by Guier and Weiffenbach,
using some of the same linearization methods used by Carl Friedrich Gauss in his 1801 determination of the o rbit
of the asteroid Ceres [2]. Gauss had far more observational data than unknowns needing to be determined. He was
then able to pose this in the f orm of an overdetermined system of equations a nd use the method of least squares
he had discovered earlier. In the case of the Doppler satelli te navigation problem, Guier and Weiffenbach could use
partial derivatives of the Doppler frequency measurements with respect to receiver location to model the antenna
location problem as an overdetermined linear system [4]. The least-squares solution o f an over-determined linear
system Hx = z for x, given H and z, is
ˆx =
H
T
H
−1
H
T
z, (C.1)
if the gramian
H
T
H
is invertibl e. The determinant of this gramian matrix then becomes a discriminant for
whether a unique solution is attainable.
C.2.2 Least Mean Squares Approach
By this time, however, least-mean-squares estimation had replaced the least-squares approach of Gauss, and the
equivalent discriminant would then be the determinant of the 2 × 2 information matrix Y , defined as
Y
def
=
X
k
H
T
k
R
−1
k
H
k
(C.2)
H
k
def
=
∂f
k,Doppler
∂E
ANT
, N
ANT
t
k
(C.3)
C.2. METHODS FOR PROVING DETERMINANCY 3
where f
k,Doppler
is the k
th
Doppler frequency measurement, the partial derivatives are with respect to the east (E)
and north (N) location of the antenna evalua ted under conditions of the Doppler measurement made at time t
k
, and
R
k
is the variance of frequency measurement uncertainty on the k
th
Doppler frequency measurement
1
.
If the determinant of the information matrix Y is non-zero, then Y is nonsingular (i.e., invertible), and its inverse
P
def
= Y
−1
(C.4)
is the 2 × 2 covariance matrix of mean-squared horizontal antenna location uncertainty after all the measurements
from one or more satell ite passes. In that case, the square root of its diagonal elements
√
p
11
+ p
22
=
q
E
hε
2
E
+ ε
2
N
i (C.5)
is the RMS horizontal (2D) estimation error of antenna location.
The information matrix from the least-mean-squares approach then provides a quantitative assessment of the
solution accuracy, whereas the determinant of the gramian had been only a qualitative indicator of the feasibility of
a unique soluti on.
C.2.3 Dilution of Prec ision (DOP) Approach
Dilution of precision, is a term initially introduced to describe the impact of relative locations of radio beacons
on ground-based radionavigation systems such as LORAN. Navigational accuracy of these phase-based or relative-
timing-based systems can be different in different local horizontal directions, depending on the relative directions of
arriva l of signals from different beacons. The same is true for timing-based satellite navigation systems such as GPS
and other global navigatio n satellite systems [3]. It is also true for D oppler-based satellite navigati on, even though
it may use a single satellite.
This general approach factors the RMS horizontal position uncertainty into the product of two factors:
1. The mean-squared measurement error R.
2. The inverse of the unscaled information matri x Y
?
, which represents the effect of the choice of the measurements
to be used (but not their mean-squared uncertainties R
k
). The measurements used in this application will be
the Doppler frequencies from a passing satellite.
The unscaled information matrix characterizes performa nce metrics for the variables being estimated. The perfor-
mance metric of interest in thi s case the horizontal dilution of precision (HDOP), which is defined as
2
HDOP
def
=
√
a
11
+ a
22
(C.6)
A =
a
11
a
12
a
21
a
22
(C.7)
def
= {Y
?
}
−1
(C.8)
Y
?
def
=
X
k
H
T
k
H
k
(C.9)
H
k
def
=
∂f
k,Doppler
∂E
ANT
, N
ANT
t
k
(C.10)
where, as before, f
k,Doppler
is the k
th
Doppler frequency measurement, the partial derivatives are with respect to
east and north displacements of antenna l ocation, and the partial derivatives are evaluated under conditions of the
Doppler measurement made at time t
k
from a specified antenna location.
The matrix Y
?
is called the unscaled info rmation matrix to distinguish it from the information matrix Y defined
in Equation C.2. Y
?
is not scaled by the inverse of the mean-squared measurement uncertainty.
1
In this preliminary analysis, R
k
will be considered time-invariant. However, some variation of measur ement accuracy with time mig ht
be expected fr om the variation is satellite signal-to-noise ratio as the satellite passed overhead from horizon to horizon.
2
Here, we follow the notation of [3] in defining HDOP.
