![Page 1: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/1.jpg)
EESC 9945
Geodesy with the Global PositioningSystem
Classes 11: Stochastic Filters II
![Page 2: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/2.jpg)
Review: Kalman Filter equations
Prediction
xk|k−1 = Sk xk−1|k−1
Λk|k−1 = Sk Λk−1|k−1 STk +RkQkR
Tk
Gain
Kk = Λk|k−1ATk
(AkΛk|k−1A
Tk +Gk
)−1
Update
xk|k = xk|k−1 +Kk∆yk
Λk|k = (I −KkAk) Λk|k−1
2
![Page 3: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/3.jpg)
Example: Random walk
• Let us take the simple 1-D example
zk+1 = zk + ξk
ξk a zero-mean white-noise process with variance σ2ξ
• zk is a random-walk process
• Let us also assume an observation
yk = zk + εk
with εk our observation error: 〈ε2k〉 = σ2y
• Then Sk = Rk = Ak = I1×1 = 1
3
![Page 4: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/4.jpg)
White Noise Process
-4
-3
-2
-1
0
1
2
3
4
Whi
te N
oise
(σ2 =
1)
10008006004002000
±1 standard deviation region shown
4
![Page 5: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/5.jpg)
Random Walk Process
-40
-30
-20
-10
0
10
20
30
40
Ran
dom
Wal
k (σ
2 = 1
)
10008006004002000
±1 standard deviation region shown
5
![Page 6: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/6.jpg)
Random walk
• Let us take the simple 1-D example
zk+1 = zk + ξk
ξk a zero-mean white-noise process with variance σ2ξ
• zk is a random-walk process
• Let us also assume an observation
yk = zk + εk
with εk our observation error: 〈ε2k〉 = σ2y
• Then Sk = Rk = Ak = I1×1 = 1
6
![Page 7: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/7.jpg)
Random walk
• The prediction is
zk|k−1 = zk−1|k−1
σ2k|k−1 = σ2
k−1|k−1 + σ2ξ
• The Kalman gain is
Kk =σ2k−1|k−1 + σ2
ξ
σ2k−1|k−1 + σ2
ξ + σ2y
7
![Page 8: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/8.jpg)
Random walk
• The update is
zk|k = zk−1|k−1+
σ2k−1|k−1 + σ2
ξ
σ2k−1|k−1 + σ2
ξ + σ2y
[yk − zk−1|k−1
]
σ2k|k =
σ2y
σ2k−1|k−1 + σ2
ξ + σ2y
σ2k−1|k−1
8
![Page 9: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/9.jpg)
Random walk
zk|k = zk−1|k−1 +
σ2k−1|k−1 + σ2
ξ
σ2k−1|k−1 + σ2
ξ + σ2y
[yk − zk−1|k−1
]
σ2k|k =
σ2y
σ2k−1|k−1 + σ2
ξ + σ2y
σ2k−1|k−1
• Suppose we have an initial “guess” of z0|0 = Z◦with a large uncertainty, so σ2
0|0 � σ2ξ , σ
2y
• Then after the first step (k = 1) we have
z1|1 ' y1 σ21|1 ' σ
2y
9
![Page 10: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/10.jpg)
Random walk
• After n steps we have
σ2n|n =
[β
(1 + β)n − 1
]σ2y , β = σ2
ξ /σ2y
• Limits:
1. β → 0 (σ2ξ → 0):
σ2n|n/σ
2y →
1
n
2. β →∞ (σ2y → 0): σ2
n|n → σ2y/β
n−1
σ2n|n/σ
2y →
1
βn−1
10
![Page 11: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/11.jpg)
Random-Walk Example
10-30 10-27 10-24 10-21 10-18 10-15 10-12 10-9 10-6 10-3 100
σ2 n|n /
σ2 y
12 3 4 5 6 7 8
102 3 4 5 6 7 8
100n
β = 0.001 β = 0.1 β = 1
11
![Page 12: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/12.jpg)
Random walk: Interpretation of results
• In this example, the state is a random walk (zk+1 =zk + ξk), and we directly observe the state (yk =zk + εk)
• Why does result for σ2k|k depend on β = σ2
ξ /σ2y?
• Recall that the derivation of the sequential least-squares used the weighted mean of the predictedstate and the state derived from the observation
• From the point of view of least-squares, each ofthese state estimates is a random variable, equiva-lent except for their standard deviations
12
![Page 13: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/13.jpg)
Random walk: Interpretation of results
• A small value for β is roughly equivalent to the
statement, The state is walking around, but nearly
imperceptibly compared to our ability to observe it
• Thus, the filter acts to mostly keep the state fixed in
time, and to ascribe any variability to observational
error
• Hence, σn|n ∼ n−1 just like a constant mean value
is being estimated
13
![Page 14: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/14.jpg)
Random walk: Interpretation of results
• A large value for β is roughly equivalent to the state-
ment, The state is walking around, and compared
to this variability we can observe it nearly perfectly
• Thus, the filter ascribes any variability to real vari-
ability of the state itself. . .
• . . .and zero uncertainty to error in the estimate of
the state
14
![Page 15: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/15.jpg)
Random walk: Interpretation of results
• The preceding interpretation shows why this is called
a filter
• The filter takes (in this example) or a random input
observations), and a stochastic model, and based
on the details of the model separates the signals
into two components:
– The estimate of the state vector xk
– The post-fit residual εk = yk −Akxk
15
![Page 16: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/16.jpg)
Kalman Filter Schematic
+ - + +K(t) unit
delayS(t)
A(t)
y(t)
y(t|t-1)
x(t|t) x(t-1|t-1) x(t|t-1)
e(t) = y(t) - A(t) x(t|t)
16
![Page 17: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/17.jpg)
1-D position with random-walk speed
Transponder
• Speed varies as random walk (wind? currents?)
• Temporal variability (ξk zero-mean white-noise, σ2ξ ):
zk = zk−1 + vk−1∆t
vk = vk−1 + ξk
• Must include both position and speed in state
17
![Page 18: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/18.jpg)
1-D position with random-walk speed
• Dynamic model for state:
xk =
[zkvk
]=
(1 ∆t0 1
)xk−1 +
(01
)ξk
• Observe two-way range delay (obs. sigma σ2y):
yk =(
2c 0
)xk + εk
• The partial derivative with respect to speed is zero!
• Can we estimate the speed?
18
![Page 19: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/19.jpg)
1-D position with random-walk speed
• Kalman gain
Kk = Λk|k−1ATk
(AkΛk|k−1A
Tk +Gk
)−1
• Since(AkΛk|k−1A
Tk +Gk
)−1is 1× 1:
Kk ∼ Λk|k−1ATk
• For Λk|k−1 =
(σ2z σzv
σzv σ2v
):
Kk ∼(σ2z
σzv
)
19
![Page 20: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/20.jpg)
1-D position with random-walk speed
• Thus, there is information to estimate both param-eters even though observation = distance
• Even on first epoch, if Λ0|0 =
(σ2z 0
0 σ2v
), then
Λ1|0 =
(σ2z + (∆t)2σ2
v σ2v∆t
σ2v∆t σ2
v + σ2ξ
)=
(σ2z σ2
v∆t
σ2v∆t σ2
v + σ2ξ
)
• And Kalman gain
K1 =(c
2
) [σ2z +
(c2
4
)σ2y
]−1(σ2z
σ2v∆t
)
20
![Page 21: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/21.jpg)
ξ (White Noise)
-4
-3
-2
-1
0
1
2
3
4
ξ
10008006004002000
21
![Page 22: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/22.jpg)
Speed (Random Walk)
40
30
20
10
0
-10
-20
Speed
10008006004002000
22
![Page 23: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/23.jpg)
Position (Integrated Random Walk)
12000
10000
8000
6000
4000
2000
0
-2000
Position
10008006004002000
23
![Page 24: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/24.jpg)
Application to GPS positioning
• Stochastic parametrization is used by various soft-ware packages for
– Clock errors
– Atmospheric (wet) zenith delay
– Zenith-delay gradient parameters
– Time-dependent positions
– Earth orientation/rotation
– Orbit parameters
24
![Page 25: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/25.jpg)
Stochastic Models in GPS
• White noise, random walk, and integrated random
walk are most used
– They are easy to implement in a Kalman filter
– Nature of variations can be tuned to understand-
ing of physical process
• Stochastic variances have to be “guesstimated,”
and sensitivity studies done
25
![Page 26: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/26.jpg)
GPS Car Navigation Example
Position of your vehicle and the navigation system’s route
26
![Page 27: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/27.jpg)
GPS Car Navigation Example
Route you want to take
27
![Page 28: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/28.jpg)
GPS Car Navigation Example
Estimated position (blue/yellow) vs. true position (red/white)
28
![Page 29: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/29.jpg)
GPS Car Navigation Example
Estimated position (blue/yellow) vs. true position (red/white)
29
![Page 30: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/30.jpg)
GPS Car Navigation Example
Estimated position (blue/yellow) vs. true position (red/white)
30
![Page 31: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/31.jpg)
GPS Car Navigation Example
Estimated position (blue/yellow) vs. true position (red/white)
31
![Page 32: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/32.jpg)
What’s happening?
• Kalman filter for position and velocity (?)
• Vehicle is assumed to be on road
• Identified of road enables prediction with small σ’s
• As prefit residual pseudoranges (yk − Akxk|k−1) be-come large, estimated position changes, but notenough to be identified as different road
• Prefit residuals at some point become large enoughthat a true position update is allowed
32
![Page 33: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/33.jpg)
Combining high-accuracy GPS andaccelerometer data for strong-motion
displacements
• Recently, Yehuda Bock (UCSD) and colleagues haveexperimented with combining GPS and accelerom-eter estimates using a Kalman filter
• There are two ways of doing this
• One is to devise a state vector including positionand acceleration, for example (for each component):
xk =
zkvkak
33
![Page 34: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/34.jpg)
Combining GPS and accelerometer data
• The transition matrix would then be
Sk =
1 ∆t 12 (∆t)2
0 1 ∆t0 0 1
• The observation vector would include both posi-tion z (from GPS) and acceleration a (from theaccelerometer)
• This approach would require understanding the statis-tics of the acceleration changes which would bemodeled as noise
34
![Page 35: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/35.jpg)
Combining GPS and accelerometer data
• The approach taken by Bock et al. is to use amodified dynamic equation
xk = Skxk−1 +Bkuk +Rkξk
• The state vector is
xk =
[zkvk
]
• The transition matrix is
Sk =
(1 ∆t0 1
)
35
![Page 36: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/36.jpg)
Combining GPS and accelerometer data
xk = Skxk−1 +Bkuk +Rkξk
• The input uk is the accelerometer reading, so that
Bk =
[12 (∆t)2
∆t
]
• This modified dynamic equation can be implemented
by modifying the prediction equation:
xk|k−1 = Skxk−1|k−1 +Bkuk
36
![Page 37: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/37.jpg)
Combining GPS and accelerometer data
• The noise matrix Qk has a rather complex form to
reflect the sampling (with error) of a continuous
quantity (acceleration)
• The GPS receiver and accelerometer have different
sampling rates
– GPS: 10 Hz
– Accelerometer: 100 Hz
37
![Page 38: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/38.jpg)
GPS vs. KF (El Mayor-Cucapah 4/4/2010)
Bock et al. (2011), BSSA, 101, 2904–2925
38
![Page 39: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/39.jpg)
Seismometer vs. KF (El Mayor-Cucapah)
Bock et al. (2011), BSSA, 101, 2904–2925
39
![Page 40: EESC 9945 Geodesy with the Global Positioning Systemjdavis/eesc_9945_L11.pdf · 2013-04-19 · Random walk: Interpretation of results In this example, the state is a random walk (zk+1](https://reader033.vdocuments.net/reader033/viewer/2022050402/5f802fdcddfef95bea05b843/html5/thumbnails/40.jpg)
Position estimates of glacier GPS sites
Nettles et al. (2008), GRL, 35, L24503
40