grace analysis - egsiem.euegsiem.eu/images/static/autumn-school/presentations/mayer-guerr.pdf · 10...

u ifg.tugraz.at

Institute of Geodesy

Graz University of Technology

S C I E N C E P A S S I O N T E C H N O L O G Y

GRACE Analysis

Torsten Mayer-Gürr

EGSIEM autumn school

2017-09-12

2 Torsten Mayer-Gürr

Gravity Recovery and Climate Experiment



Distance: ca. 250 km



Distance: ca. 250.000000000 km

Accuracy: 1μm = 12 digits


Groundtracks


Groundtracks

1 Day

15 Days

30 Days


GRACE Monthly solutions


ITSG-Grace2016 daily (30 day smoothing)



More than 3.000 Publications

(GRACE Tellus, grace.jpl.nasa.gov,

from March 2017)

Sumatra–Andaman

earthquake 2004

Han et al., 2006, Science

Ice mass variations in Greenland

Velicogna, 2009, Geophys. Res. Lett.

Groundwater depletion in North India

Rodell et al., 2009, Nature



More than 3.000 Publications

(GRACE Tellus, grace.jpl.nasa.gov,

from March 2017)

Sumatra–Andaman

earthquake 2004

Han et al., 2006, Science

Ice mass variations in Greenland

Velicogna, 2009, Geophys. Res. Lett.

Groundwater depletion in North India

Rodell et al., 2009, Nature

Global Sea level change

IPCC WGI Fifth Assessment Report, 2014

Torsten Mayer-Gürr11

Data bases


Data base

General information

GRACE tellus: grace.jpl.nasa.gov

- Information

- Publications

- Data

International Centre for Global Earth Models (ICGEM) icgem.gfz-potsdam.de/ICGEM/

EGSIEM egsiem.eu

Processing centers

GFZ isdc.gfz-potsdam.de

CSR csr.utexas.edu/grace

JPL podaac.jpl.nasa.gov/gravity/grace

CNES: grgs.obs-mip.fr/grace

Uni Bern aiub.unibe.ch

TU Graz ifg.tugraz.at

Tongji University

Wuhan University

TU Delft

…

Combination

EGSIEM


ICGEM


Gravity field solutionproduct_type gravity_field

modelname ITG-Grace03

comment static field from 2002-08 to 2007-04 of GRACE data

earth_gravity_constant 3.986004415e+14

radius 6378136.6

max_degree 180

key n m C S sigma C sigma S

end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...

Spherical harmonics ?

Where is the water storage?


Outline

Approximation of functions on the sphere

Spherical harmonics

Gravity field

Upward continuation

Orders of magnitude

Functionals of the gravity field

Accuracy of gravity fields

Degree variances

Filtering

Signal content

Physical interpretation of the coefficients

Evaluation of GRACE solutions: Step by Step


Approximation of functions on the sphere


Approximation

Approximation with a polynomial of degree n:

)()()()( 1100 xpaxpaxpaxf nn

n

n xxp )(

Approximation of a periodic function with a Fourier series:

1

0

2sin

2cos)(

n

nn tT

mstT

mcctf

0

),(),(n

n

nm

nmnmYaf

Approximation of a function on the sphere with spherical harmonics:

x

y

coefficients (scale)degree n order m

basis functions


Coefficient triangle

Approximation of a function on the sphere

0

),(),(n

n

nm

nmnmYaf

degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m

Arrangement of coefficients in a triangle






radius 6378136.6

max_degree 180


end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...

degree






radius 6378136.6

max_degree 180


end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...

order






radius 6378136.6

max_degree 180


end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...

coefficients






radius 6378136.6

max_degree 180


end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...

accuracies


Approximation of functions on

the sphere


Approximation with spherical harmonics

Degree nNumber of

coefficients

4 25

8 81

16 289

30 961

60 3721

120 14641

240 58081

0

)()(n

n

nm

nmnmYaf xx



0

)()(n

n

nm

nmnmYaf xx

Degree nNumber of

coefficients

4 25

8 81

16 289

30 961

60 3721

120 14641

240 58081


The basis functions


The basis functions

Basis functions

)(cos)cos(),(, nmmn PmY

)(cos)sin(),(, nmmn PmY


0

),(),(n

n

nm

nmnmYaf

coefficients (scale) Basis functions

depends on longitude:

Fourier series along a

parallel of latitude

depends on latitude:

Legendre functions along a

meridian


The basis functions

Basis functions

)(cos)cos(),(, nmmn PmY

)(cos)sin(),(, nmmn PmY


0

),(),(n

n

nm

nmnmYaf

coefficients (scale) Basis functions


0 0

)(cos)sin()cos(),(n

n

m

nmnmnm PmSmCf

coefficients (scale)


Computation of the basis functions

1,1 Y0,1Y 1,1Y

2,2 Y1,2 Y 0,2Y

3,3 Y2,3 Y 1,3 Y

4,4 Y3,4 Y 2,4 Y

0,0Y

1,2Y 2,2Y

0,3Y 1,3Y 2,3Y 3,3Y

1,4 Y 0,4Y 1,4Y 2,4Y 3,4Y 4,4Y

Coordinates on sphere

cos

sinsin

sincos

z

y

x

1. 10,0 Y

2.

)(

)(

1,11,1,

1,11,1,

nnnnnnn

nnnnnnn

xYyYaY

yYxYaY

4.

mnmnmnmnmn YcYzbY ,2||,,1||,,

,2,nfor m

3.

mnmnmn YzbY ,1||,,

1for mnFactors (normalized)

31 an

nan

2

12

))((

)12)(12(

mnmn

nnbnm

))()(32(

)1)(1)(12(

mnmnn

mnmnncnm


Basis functions

),(3,4 S

),(5,20 C

),(20,40 C


Basis functions

C00

A00 C11S11 C10

C20

C30

C40

S22 S21

S31S33 S32

S41S42S44 S43

C21 C22

C31

C41

C32 C33

C42 C43 C44

C50S51S52S54 S53 C51 C52 C53 C54 C55S55

zonalsektoriell sektorielltesseral tesseral

Degree n = 0

Degree n = 1

Degree n = 2

Degree n = 3

Degree n = 4

Degree n = 5

Rummel

mPY m

nmn cos)(cos),(,

mPY m

nmn sin)(cos),(,




0

),(),(n

n

nm

nmnmYaf

degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m



Accuracies of coefficients



Accurate at low degrees

(long wave lengths)

Inaccurate at high degrees

(fine structure)



Accurate at low orders

(zonals)

Inaccurate at high orders

(sectoriells)Inaccurate at high orders

(sectoriells)






radius 6378136.6

max_degree 180


end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e-13

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e-13

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e-13

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e-13

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e-13

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e-13

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e-13

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e-13

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e-13

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e-13

...


Gravity field


Gravity field

Gravity field:

2

m

s

)(

)(

)(

)(

r

r

r

rg

z

y

x

g

g

g


Gravity field

Gravity field:

2

m

s

)(

)(

)(

)(

r

r

r

rg

z

y

x

g

g

g

Conservative vector field

Potential function exists:

zV

yV

xV

V )()( rrg )(rV

2

2

s

m

Source free in outer space

0div gg

0 VV

Laplace equation (harmonic func.)

02

2

2

2

2

2

z

V

y

V

x

VV


Harmonics continuation

Potential at Earth‘s surface

),( V

Harmonic continuation

02

2

2

2

2

2

z

V

y

V

x

VV

Potential in outer space

dVl

rRRrV ),(

1

4),,(

3

22

cos222 RrrRl with


Harmonics continuation

Potential at Earth‘s surface

(in spherical harmonics)

0

),(),(n

n

nm

nmnmYaR

GMV

Potential in outer space

(solid spherical harmonics)

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrV

Harmonic continuation

02

2

2

2

2

2

z

V

y

V

x

VV


Orders of magnitude


Gravity

Gravity at Earth surface

281,9

s

mg


Gravity


2283,978,9

s

m

s

mg


Gravity


2283,978,9

s

m

s

mg

20004,0

s

m

200001,0mGal1

s

m


Gravity anomalies

GOCE TIM5


GRACE Monthly solutions


GRACE gravity field March 2008


GRACE gravity field September 2008


September – March 2008


Gravity field

Flattening

Sphere

Static part

mGal100001.02

s

mg

Time variable part

mGal01.00000001.02

s

mg

100 C

-0.000484220 C

2210

s

m

R

GMg

203.0

s

mg

Normal field

(Geodetic Reference

System, GRS80)

Spherical

approximations +

ellipsoidal corrections

Spherical

approximations


Functionals of the gravity field


Gravity field functionals

Disturbing potential

UVT

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrT

0

),(),(n

n

nm

nmnmYaRN

gravity

Vg

Geoid heights

g

T

V

TdrN

drVVV )()( 0rr

Geoid heights changes

g

TN

Spherical approx.

2R

GMg

Normal potential


Gravity anomalies

GOCE TIM5


Sea surface at rest

GOCE TIM5


Geoid = equipotential surface

GOCE TIM5




UVT

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrT

0

),(),(n

n

nm

nmnmYaRN

gravity

Vg

Geoid heights

g

T

V

TdrN

drVVV )()( 0rr

Geoid heights changes

g

TN

Spherical approx.

2R

GMg

Normal potential



Gravity disturbances

r

TTg

0

1

),(1

),,(n

n

nm

nmnm

n

Yar

R

r

n

R

GMrg

Gravity anomlies

r

T

r

Trrgg QP 2)()(

0

1

),(1

),,(n

n

nm

nmnm

n

Yar

R

r

n

R

GMrg

Surface density changes

Rrr

T

r

T

RG

2

4

1

Total water storage changes

0'

),(1

12

3),(

n

n

nm

nmnm

n

e Yak

nRTWS


UVT

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrT

0

2),()12(

4),(

n

n

nm

nmnmYanR

M

334 R

Me

Normal potential


Accuracy of gravity fields

and

Degree variances


ITSG-Champ2011 - Geoid

How accurate is the

estimation of the geoid?

Comparison with a more

accurate reference model


ITSG-Champ2011 – GOCO05s

Geoid differences up to degree n=120

RMS = 57 cm

),(),(),( GOCOCHAMP NNN


ITSG-Champ2011 – GOCO05s

Geoid differences up to degree n=60

RMS = 1.3 cm



Variance

Geoid differences


Variance (first try):

n

i

iiNn 1

22 ),(1

with area weights:

n

i

iiiN1

22 ),(4

1

problem: a lot of values at the poles

increase density of points to infinity

Integral:

dN ),(4

1 22


Variance

Geoid differences


dN ),(4

1 22

0

),(n

n

nm

nmnmYaR

2R

GM

00

00

),(

),(

n

n

nm

nm

GOCO

nm

n

n

nm

nm

CHAMP

nm

YaR

GM

YaR

GM

00

),(n

n

nm

nmnmYaR

GM


Variance

Geoid differences

dN ),(4

1 22

0

),(n

n

nm

nmnmYaR

2R

GM


00

00

),(

),(

n

n

nm

nm

GOCO

nm

n

n

nm

nm

CHAMP

nm

YaR

GM

YaR

GM

00

),(n

n

nm

nmnmYaR

GM

dYa

R

n

n

nm

nmnm

2

0

2

),(4

dYaYa

R

n

n

nm

mnmn

n

n

nm

nmnm

0'

'

'

''''

0

2

),(),(4

dYYaaR

nmmn

n

n

nm n

n

nm

mnnm ),(),(4

''

0 0'

'

'

''

2

0

22

n

n

nm

nmaR

degree variance

n

nm

nmn a22

spherical harmonics are orthogonal

'', mnnm YY

dYY mnnm ),(),( ''

''4 m

m

n

n


Variance

dN ),(4

1 22

0

22

n

nR

n

nm

nmn a22with the degree

variance

0

22

n

nR

Root Mean Square (RMS)


RMS (accumulated up to degree N)

RMS Geoid

up to degree N

N

n

nR0

2

n

nm

nmn a22

with

RMS = 57 cm

RMS = 1.3 cm


RMS (degree N)

RMS Geoid

of degree N

n

nm

nmn a22

with

2

nR


RMS (degree N)

RMS Geoid

of degree N

n

nm

nmn a22

with

2

nR

Formal errors

(from adjustment)

2ˆˆnR

n

nm

nmn a )(ˆˆ 22

with


Signal content

Difference degree

variances

Variance of geoid differences

dNN ),(4

1)( 22

0

22

n

nR

n

nm

nmn a22

Variance/Variability of geoid heights

dNN ),(4

1)( 22

0

22

n

nR

Degree variances

n

nm

nmn a22


RMS (degree N)

RMS Geoid

of degree N

2

nR

n

nm

nmn a22

mit

Formal errors

(from adjustment)

2ˆˆnR

with

n

nm

nmn a )(ˆˆ 22


RMS (degree N)

RMS Geoid

of degree N

2

nR

n

nm

nmn a22

mit

Formal errors

(from adjustment)

2ˆˆnR

with

n

nm

nmn a )(ˆˆ 22

Signal2

nR

n

nm

nmn a22

with


Upward continuation


Upward continuation

Gravitational potential

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrV

0 km600 km 20000 km


Upward continuation

0

1

),(),,(n

n

nm

nmnm

n

Yar

R

R

GMrV

180für0.000004

120für0.000261

2für0.815029

0für934095,0

1

1

1

1

nr

R

nr

R

nr

R

nr

R

n

n

n

n

kmRr

kmR

450

6378

(110 km)

(160 km)

(10.000 km)

Gravitational potential

GRACE Damping factors


Degree variances

Satellite height (450 km):

: Earth surface (0 km)


Degree variances




Degree variances

Signal

Accuracy of

monthly solution


Degree variances

Signal

Accuracy of

monthly solution

Difference of two

monthly solution


Degree variances

Signal

Accuracy of

monthly solution

Difference of two

monthly solution

Time variations


Filtering


Filtering

Filtering the gravity field (Convolution)

)()(),(4

1)( yyyxx dVFVF

Gaussian Filter F Gaussian filter

)cos1(

21

2)(cos

b

be

e

bF

)cos(1

)2ln(

Rd

b

with

with d: filter radius


Degree variances

Remove the errors in higher degrees (green line)

Multiply the coefficients of the higher degrees with factor 0

Preserve the time variable part (black line)

Multiply the coefficients of the lower degrees with factor 1

Gaussian filter:

smooth transition between low and high degrees


Gaussian filter


Gaussian filter

0

1

),(),(),,(n

n

nm

nmnmnmnmn

n

F SsCcwr

R

R

GMrV

Filtered potential

Gaussian filter coefficients

be

ew

b

b 1

1

12

2

1

10 w

21

12

nnn ww

b

nw

with

)cos(1

)2ln(

Rd

b


200 km … 1000 km


Filtering

Filtering in the spectral domain

0

)()(n

n

nmnmnmnF Yaw

R

GMV xx

Filter coefficients

be

ew

b

b 1

1

12

2

1

10 w

21

12

nnn ww

b

nw

Unfiltered potential

0

)()(n

n

nmnmnmYa

R

GMV yy

Filtering in the spatial domain

)()(),(4

1)( yyyxx dVFVF

Gaussian Filter

)cos1(

21

2)(cos

b

be

e

bF


Physical interpretation

of the coefficients




0

),(),(n

n

nm

nmnmYaf

degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m





0

),(),(n

n

nm

nmnmYaf


degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m

Total mass,

Sphere




0

),(),(n

n

nm

nmnmYaf


degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m

Center of mass

of complete Earth


Geocenter motion

R. Rietbroek

GRACE orbits around the center of total mass

(Fluids (bycycle) + Solid Earth)

Degree 1 coefficients are always zero

To analyize data in an Earth fixed system,

external time series of degree 1 coefficients

are needed.

GRACEEarth fixed

e.g.:

www.igg.uni-bonn.de/apmg/index.php?id=geozentrum




0

),(),(n

n

nm

nmnmYaf


degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m

Center of mass

of complete Earth




0

),(),(n

n

nm

nmnmYaf


degree n = 0

5,5 a4,5 a 3,5 a 2,5 a 1,5 a

50a 51a 52a 53a 54a 55a

4,4 a 3,4 a 2,4 a 1,4 a40a

41a 42a 43a44a

3,3 a 2,3 a 1,3 a30a 31a 32a 33a

2,2 a 1,2 a20a

21a 22a

1,1 a10a

11a

00a

degree n = 1

degree n = 2

degree n = 3

degree n = 4

…

order m

Flattening of the

Earth


Signal content


Signal content

All masses causes gravity

GRACE cannot separate the different signal sources

• Hydrology

• Ice sheets

• Glaciers

• Permanent frost

• Ocean tides

• Ocean pole tides

• Barotropic ocean circulation

• Sea level rise

• Atmospheric tides (S1, S2)

• Atmospheric mass redistribution

• Solid Earth tides

• Rotational deformation (pole tides)

• Glacial isostatic adjustment

• Loading deformation

• Degree 1 mass redistribution

• Earthquakes

Ice

Ocean

Atmosphere

Solid Earth


Signal content

All masses causes gravity

GRACE cannot separate the different signal sources

• Hydrology

• Ice sheets

• Glaciers

• Permanent frost

• Ocean tides

• Ocean pole tides

• Barotropic ocean circulation

• Sea level rise

• Atmospheric tides (S1, S2)

• Atmospheric mass redistribution

• Solid Earth tides

• Rotational deformation (pole tides)

• Glacial isostatic adjustment

• Loading deformation

• Degree 1 mass redistribution

• Earthquakes

Some of the signals are not included

in the GRACE solutions,

but directly reduced by models


Evaluation of GRACE solutions

Step by Step


Step by Step

1. Download the spherical harmonics coefficients

product_type gravity_field




radius 6378136.6

max_degree 180

key n m C S sigma C sigma

end_of_head =========================================================================

gfc 0 0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 0 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 1 1 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00 0.000000000000e+00

gfc 2 0 -4.841692718699e-04 0.000000000000e+00 6.469883774458e-13 0.000000000000e+00

gfc 2 1 -2.654790999243e-10 1.475393314283e-09 6.108979511966e-13 6.355307212507e

gfc 2 2 2.439383367978e-06 -1.400273635220e-06 6.254221806143e-13 6.423410956098e

gfc 3 0 9.571610348416e-07 0.000000000000e+00 4.908157850872e-13 0.000000000000e+00

gfc 3 1 2.030461736678e-06 2.482003394707e-07 4.904543816334e-13 5.118675157415e

gfc 3 2 9.047877724984e-07 -6.190053685183e-07 5.459595906001e-13 5.482674767117e

gfc 3 3 7.213217237276e-07 1.414349090196e-06 5.163836126113e-13 5.163483433061e

gfc 4 0 5.399657665980e-07 0.000000000000e+00 3.758481731782e-13 0.000000000000e+00

gfc 4 1 -5.361573220519e-07 -4.735672404588e-07 3.874699557956e-13 3.973550735355e

gfc 4 2 3.505015650151e-07 6.624798955603e-07 4.501829959710e-13 4.398486563277e

gfc 4 3 9.908565738322e-07 -2.009566568843e-07 4.776067657084e-13 4.766590461153e

gfc 4 4 -1.885196275153e-07 3.088038091544e-07 4.556511148108e-13 4.565287154679e

gfc 5 0 6.867029195170e-08 0.000000000000e+00 2.444626602968e-13 0.000000000000e+00

gfc 5 1 -6.292117216708e-08 -9.436975416042e-08 2.510438328896e-13 2.640391465060e

...


Step by Step


2. Insert external coefficients of degree 1

R. Rietbroek

GRACEEarth fixed


Step by Step



3. Remove static part of the gravity field

(temporal mean of time series

or other static field e.g. GOCO05s)

0

nmnmnm aaa


Step by Step






4. Remove unwanted signals, e.g. GIA

0

nmnmnm aaa


Step by Step







5. Filter the coefficients (e.g. Gaussian filter)

0

nmnmnm aaa

nmnnm awa

Filter coefficients

be

ew

b

b 1

1

12

2

1

10 w

21

12

nnn ww

b

nw

)cos(1

)2ln(

Rd

b

with



Step by Step







5. Filter the coefficients (e.g. Gaussian filter)

6. Compute Total water storage changes (TWS)

at each grid point

0

nmnmnm aaa

nmnnm awa

),( ii

0'

),(1

12

3),(

n

n

nm

nmnm

n

e Yak

nRTWS

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