geodesy esci7355f 2010 class 8

7/27/2019 Geodesy ESCI7355f 2010 Class 8

1/82

Earth Science Applications of Space Based GeodesyDES-7355

Tu-Th 9:40-11:05Seminar Room in 3892 Central Ave. (Long building)Bob SmalleyOffice: 3892 Central Ave, Room 103678-4929Office Hours Wed 14:00-16:00 or if Im in my office.

http://www.ceri.memphis.edu/people/smalley/ESCI7355/ESCI_7355_Applications_of_Space_Based_Geodesy.html

Class 8 1


2/82

2

If X is a continuous random variable, thenthe probability density function, pdf, of X,

is a function f(x) such that for two numbers, a and b with ab

P a x b fa

b

x dx

That is, the probability that X takes on a value in theinterval [a, b] is the area under the density function from a tob.http://www.weibull.com/LifeDataWeb/the_probability_density_and_cumulative_distribution_functions.htm


3/82

3

The probability density function for the Gaussiandistribution is defined as:

From G. Mattioli

PG x,, 1

2exp

1

2

x

2


4/82

4

For the Gaussian PDF, the probability for the randomvariable x to be found between z,Where z is the dimensionless range z = |x -|/ is:

From G. Mattioli

AG x,, PGz

z

x,, dx 12

exp 1

2x 2

dx

z

z

AG z 1


5/82

5

The cumulative distribution function, cdf,is a function F(x) of a random variable, X, andis defined for a number x by:

F x P X x f0,

x

s ds

That is, for a given value x, F(x) is the probability that theobserved value of X will be at most x. (note lower limit showsdomain of s, integral goes from 0 to x


6/82

Relationship between PDF and CDFDensity vs. Distribution Functions for Gaussian

-> integral ->


7/82

7

Expected value or mean of sum of two random variables issum of the means.known as additive law of expectation.

E x y E x E y

Multiple random variables


8/82

8

xy2 COV x,y 1

n 1 xi E x yi E y

i1

n

xy2 COV x,y pxyi xi E x yi E y

i1

n

covariance

(variance is covariance of variable with itself)(more general with) individual probabilities


9/82


10/82

10

Covariance matrix defines error ellipse.Eigenvalues are squares of semimajor and semiminor axes(1 and 2)

Eigenvectors give orientation of error ellipse(or given x and y, correlation gives fatness and angle)


11/82

11

Distance Root Mean Square (DRMS, 2-D extension ofRMS)

RMS 1n

x i2

i1

n

DRMS x2 y2

12

For a scalar random variable or measurement with a Normal(Gaussian) distribution,the probability of being within the 1- ellipse about themean is 68.3%

Etc for 3-D


12/82

12

common practice to use the reciprocal of the variance as theweight

Use of variance, covariance in Weighted Least Squares


13/82


14/82

14ttp://www.kaspercpa.com/statisticalreview.htm

Multiplying a random variable by a constant increases thevariance by the square of the constant.

cx2 E cx c2E x


15/82

15

CorrelationThe more tightly the points are clustered together thehigher the correlation between the two variables and thehigher the ability to predict one variable from another

Ender, http://www.gseis.ucla.edu/courses/ed230bc1/notes1/var1.html

y=mx+b

y=?(x)


16/82

16

Correlation coefficients are between -1 and +1,+ and - 1 represent perfect correlations,

and zero representing no relationship, between thevariables.


y=mx+b

y=?(x)


17/82

17

Correlations are interpreted by squaring the value of thecorrelation coefficient.The squared value represents the proportion of variance ofone variable that is shared with the other variable,

in other words, the proportion of the variance of onevariable that can be predicted from the other variable.



18/82

18

Sources of misleading correlation(and problems with least squares inversion)

outliersBimodaldistribution Norelation


19/82

19

Sources of misleading correlation(and problems with least squares inversion)

Restriction of range

Combininggroups

curvelinearity


20/82

20

rule of thumb for interpreting correlation coefficients:Corr Interpretation

0 to .1 trivial.1 to .3 small.3 to .5 moderate.5 to .7 large.7 to .9 very large



21/82

21

Correlations express the inter-dependence betweenvariables.For two variables x and y in a linear relationship, thecorrelation between them is defined as

xy xy

xy

http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap7/725.htm


22/82

22

High correlation does not mean that the variations of oneare caused by the variations of the others, although it maybe the case.

two types of correlation

In many cases, external influences may be affecting bothvariables in a similar fashion.

physical correlation and mathematical correlationhttp://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap7/725.htm


23/82


24/82

24

Mathematical correlation is related to the parameters in themathematical model.It can therefore be partitioned into two further classeswhich correspond to the two components of themathematical adjustment model:

Functional correlationStochastic correlation


25/82

25

Functional Correlation:The physical correlations can be taken into account byintroducing appropriate terms into the functional model ofthe observations.

That is, functionally correlated quantities share the sameparameter in the observation model.An example is the clock error parameter in the one-wayGPS observation model, used to account for the physicalcorrelation introduced into the measurements by thereceiver clock and/or satellite clock errors.


26/82

26

Stochastic Correlation:Stochastic correlation (or statistical correlation) occurs

between observations when non-zero off-diagonal elementsare present in the variance-covariance (VCV) matrix of theobservations.Also appears when functions of the observations areconsidered (eg. differencing), due to theLaw of Propagation of Variances.

However, even if the VCV matrix of the observations isdiagonal (no stochastic correlation), the VC V m atrix of theresultant LS estimates of the parameters will generally befull matrices, and therefore exhibit stochastic correlation.


27/82

27

Covariance and Cofactor matrix in GPSIf observations had no errors and the model was perfectthen the estimations from

x ATA

1AT

b

Would be perfect


28/82


29/82

29

If we have an expected (a priori) value for the error in thedata, , we can compute the expected error in theparametersConsider the covariance matrix

and for this discussion suppose that the observations areuncorrelated (covariance matrix is therefore diagonal)Cii Ei2 i

2

C ET


30/82

30

Cii

2I

Assume further that we can characterize the error in theobservations by a single number, .

Cx ExxT E ATA 1ATr ATA 1ATr

T

Cx E ATA 1ATrTA ATA 1

Cx ATA 1ATE rT A ATA 1

Cx ATA

1AT2IA ATA

1

Cx 2 ATA 1 ATA ATA 1

Cx 2 ATA 1

then


31/82

31

Cx 2 ATA 1

r

x AT

A

1

A

Tr

x ATA 1ATr

b

Expected covariance is 2 (a number) times cofactormatrix, same form as

Covariance or cofactor matrixATA

1


32/82

32

Interpretation of covariance

Cx 2 ATA

1

Variance of measurementsWe saw before that A isdependent on the directionfrom antenna to satellite so

it is a function of thegeometry

Measurement errors maybe independent (ourassumption why we couldfactor out constant 2 )But total effect, after Least Squares, can be non-

diagonal.


33/82

33

ATA 1

Since A is function of geometry only, the cofactor matrix isalso a function of geometry only.

Can use cofactor matrix to quantify the relative strength ofthe geometry.Also relates measurement errors to expected errors inposition estimations


34/82

34

In the old days,before the full constellation of satellites was flying,one had to plan design the GPS surveying sessionsbased on the (changing) geometry.

A is therefore called the design matrixDont have to worry about this anymore

(most of the time).


35/82

35

Look at full covariance matrix

Cx 2 ATA 1

Cx 2

x2 xy xz x

yx y2 yz y

zx zy z2 z

x y z 2

ij jiOff diagonal elements indicate degree of correlationbetween parameters.


36/82

36

Correlation coefficient

ij ij

i2

j2

Depends only on cofactor matrixIndependent of observation variance (the 2 s cancel out)

+1 perfect correlation what does it mean the twoparameters behave practically identically (and notindependent?!).0 no correlation, independent

-1 perfect anti-correlation practically opposites (and notindependent)


37/82

37

So far all well and good but Cartesian coordinates are not the most useful.We usually need estimates of horizontal and verticalpositions on earth (ellipsoid?).

We also need error estimates on the position.Since the errors are a function of the geometry only, onemight expect that the vertical errors are larger than the

horizontal errors.


38/82

38

How do we find the covariance / cofactor matrices in thelocal (north, east, up) coordinate system?

Have to transform the matrixfrom its representation in one coordinate systemto its representation in anotherusing the rules of error propagation.


39/82

39

First how do we transform a small relative vector inCartesian coordinates (u,v,w)

to local topocentric coordinates (n,e,u)?

L GXneu

sincos sinsin cos

sin cos 0

coscos cossin sin

xyz

Where f and are the lat and long of the location (usuallyon the surface of the earth) respectively


40/82

40

Errors (small magnitude vectors) transform the same way

G

x

Why?

L G

X

A(r

Xrx ) G

r

X Grx

r

L Grx

r

L rL

linearity


41/82


42/82

42

CL EL

L

T CL E G rx G rx

T

CL E G

rx

rx

TGT

CL GEr

xr

xT

GT

CL GCxGT

law of propagation of errors

(does this look familiar?)


43/82


44/82

44

An affine transformation is any transformation thatpreservescollinearity(i.e., all points lying on a line initially still lie on a line aftertransformation)

and ratios of distances(e.g., the midpoint of a line segment remains the midpointafter transformation).http://mathworld.wolfram.com/AffineTransformation.html


45/82

45http://mathworld.wolfram.com/AffineTransformation.html

Geometric contraction, expansion, dilation, reflection,rotation, shear, similarity transformations, spiral similarities,and translationare all affine transformations,as are their combinations.

In general, an affine transformation is a composition ofrotations, translations, dilations, and shears.

While an affine transformation preserves proportions onlines, it does not necessarily preserve angles or lengths.


46/82

46

Look at full covariance matrix(actually only the spatial part)

Cx 2 ATA 1

CL 2n

2

ne nh

en e2 eh

hn he h2

Can use this to plot error ellipses on a map (horizontalplane).


47/82

47


48/82

48

Error estimators Remember the expression for the RMS in 2-D from before

DRMS x2 y2 1

2

We can now apply this to the covariance matrix

Error estimates called dilution of Precision DOP are


49/82

49

defined in terms of the diagonal elements of the covariancematrix

GDOP n2 e2 h2 2

12

PDOP n2 e2 h2

1

2

HDOP n2 e2

1

2

VDOP hTDOP

G geometric, P position, H horizontal, V vertical,T - time


50/82

50

The DOPs map the errors of the observations(represented/quantified statistically by thestandard deviations)Into the parameter estimate errorsGDOP n

2e

2h

2

2

1

2

PDOP n2 e2 h2 1

2

HDOP n2 e2 1

2

VDOP hTDOP


51/82

51

So for a of 1 m and an DOP of 5, for example,errors in the position (where is one of G, P, H, V, T)

Would be 5=5 mGood geometry gives small DOPBad geometry gives large DOP(it is relative, but PDOP>5 is considered poor)


52/82

52ww.eng.auburn.edu/department/an/Teaching/BSEN_6220/GPS/Lecture%20Notes/Carrier_Phase_GPS.pdf


53/82

53

In 2-DThere is a 40% chance of being inside the 1- error ellipse(compared to 68% in 1-D)

Normally show 95% confidence ellipses, is 2.54 s in 2-D(is only 2 in 1-D)Can extend to 3-D


54/82

54

Another method of estimating locationPhase comparison/Interferometer

- VLBI- GPS-Carrier Phase Observable

VLBI


55/82

55http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html

Uses techniques/physics similar to GPS but with naturalsources(in same frequency band and suffers from similar errors)


56/82

56

Correlate signal at two (or more) sites to find time shiftNeed more than 1 receiver.

Differential (difference) method(similar to PRN correlation with GPS codes

orAligning two seismograms that are almost same but havetime shift)


57/82

57

Assume you are receiving a plane wave from a distantquasar

http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html Two radio antennas observe signal from quasar


58/82

58

simultaneously.The signal arrives at the two antennae at different times

The distance or baseline length b between the two
http://www.colorado.edu/engineering/ASEN/asen5090/fairbanks_antenna.gifhttp://www.colorado.edu/engineering/ASEN/asen5090/fairbanks_antenna.gif


59/82

59

antennascan be defined as:b * cos (q) = c * T where q is the angle between the baseline and the quasar

q

http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html

T
http://www.colorado.edu/engineering/ASEN/asen5090/fairbanks_antenna.gifhttp://www.colorado.edu/engineering/ASEN/asen5090/fairbanks_antenna.gif


60/82

60

Baseline lengthMassachusetts to Germany

What is this variation?Seasonal variation, geophysical phenomena, modelingproblems?http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html


61/82

61

Short period (hours/days) variations in LOD

Mostly from ocean tides and currentshttp://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html


62/82

62

Correlation of Atmospheric Angular Momentum with(longer period weeks/months) variations in LOD.



63/82

63http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html

(longer term months/years/) changes in LOD andEOP

Exchange angular momentum between large earthstructures (eg. core!) and Moon, Sun.


64/82

64

Plate velocities



65/82

65

VLBI

Not the most portable or inexpensive system But best definition of inertial reference frame external to

earth.Use to measure changes in LOD and EOP due togravitational forces and redistribution of angular momentum.http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html


66/82

66

vertical cut - spectral decomposition LOD at that instant.horizontal cut - how strength of component varies with time.Combining both 2-D view dynamic nature of LOD.



67/82

67

dominant featuresmonthly and half-monthly lunar tides, the ~800-day quasi-biennial oscillation, and the ~1600-day El Nino (dark redstructure in 1983). Yearly and half-yearly seasonalexcitations caused by meteorological variations have been

removed for clarity.http://www.colorado.edu/engineering/ASEN/asen5090/asen5090.html

dark red peaks

dark blue -troughs.

F t ff ti g EOP


68/82

68

Factors affecting EOP

lambeck-verheijen

Great book


69/82

69

G

Longitude, by Dava Sobel,describes one of the first great scientific competitions--to provide ship captains with their position at sea.

This was after the loss of two thousand men in 1707, whenBritish warships ran aground entering the EnglishChannel.The competition was between the Astronomers Royal,using distance to the moon and its angle with the stars, anda man named John Harrison, who made clocks.

Strang, http://www.siam.org/siamnews/general/gps.htm The accuracy demanded in the 18th century was a modest


70/82

70

1/2 degree in longitude.The earth rotates that much in two minutes.

For a six-week voyage this allows a clock error of threeseconds per day.Newton recommended the moon, and a German namedMayer won 3000 English pounds for his lunar tables. Even

Euler got 300 for providing the right equations.But lunar angles had to be measured, on a rolling ship atsea, within 1.5 minutes of arc.Strang, http://www.siam.org/siamnews/general/gps.htm


71/82

71

The big prize was practically in sight, when Harrison camefrom nowhere and built clocks that could do better.(You can see the clocks at GreenwichCompeting in the long trip to Jamaica, Harrisons clocklost only five seconds and eventually won the prize.

Strang, http://www.siam.org/siamnews/general/gps.htm


72/82

72

The modern version of this same competition was betweenVLBI and GPS.Very Long Baseline Interferometry uses Gods satellites,the distant quasars.

The clock at the receiver has to be very accurate andexpensive.The equipment can be moved on a flatbed, but it is certainlynot handheld.



73/82

73

There are valuable applications of VLBI, but it is GPSthat will appear everywhere.

GPS is perhaps the second most important militarycontribution to civilian science, after the Internet.

The key is the atomic clock in the satellite, designed byuniversity physicists to confirm Einsteins prediction of thegravitational red shift.Strang, http://www.siam.org/siamnews/general/gps.htm


74/82

74

Using pseudo-range, the receiver solves a nonlinear problemin geometry.

What it knows is the difference d ijbetween its distances tosatellite i and to satellite j.



75/82

75

In a plane (2-D), when we know the difference d12 betweenthe distances to two points, the receiver is located on ahyperbola.

In space (3-D) this becomes a hyperboloid.

Strang, http://www.siam.org/siamnews/general/gps.htm GPS Concepts -- 3DSoftware.com_file


76/82

76

Then the receiver lies at the intersection of threehyperboloids, determined by d12, d13, and d14.Two hyperboloids are likely to intersect in a simple closed

curve.The third probably cuts that curve at two points. But again,one point is near the earth and the other is far away.



77/82

77ttp://www.space.com/scienceastronomy/astronomy/interferometry_101.html

InterferometerBased on interference of waves


78/82

78ttp://www.space.com/scienceastronomy/astronomy/interferometry_101.html

How to make principle of interference useful?i.e. how does one get relative phase difference to vary, sothe interference varies?

Interference from single slit


79/82

79ttp://badger.physics.wisc.edu/lab/manual2/node17.html

As move across screen get phase difference from differentlengths of paths through slits

Makes fringesAs phase goes through change of 2 Interference from double (multiple) slit


80/82

80ttp://badger.physics.wisc.edu/lab/manual2/node17.html

Similar for multi-slits, but now interference is between thewaves leaving each slit

Light going through slits has to be coherent(does not work with white light)


81/82

81

The phase change comes fromthe change in geometric lengthbetween the two rays

(change in length of 1/2 wavelength causes change inphase and destructive interference)

Michelson Interferometer


82/82

Make two paths from same source(for coherence, cant do with white light)

Can change geometric path length with movable mirror.

geodesy esci7355f 2010 class 8

Documents