4.74.8

4.95

5.15.2

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0

2000

4000

6000

8000

GPS data LOS ascending

50 100 150 200 250 300 350 400

50

100

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200

250

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450 0

0.01

0.02

0.03

0.04

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0.06

=+4.8

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3D Displacement vectors

Simultaneous and Integrated Strain Tensor Estimation from geodetic and satellite deformation Measurements

A new global approach to obtain three-dimensional displacement maps by integrating GPS and DInSAR data

F. Guglielmino1, G. Nunnari2, G. Puglisi1, A. Spata2

1Istituto Nazionale di Geofisica e Vulcanologia Sez. di Catania, Italy2Department of Electrical, Electronic and System Engineering, University of Catania, Italy

The SISTEM method

DInSAR dataGlobal Positioning System

• High temporal resolution• Puntual measure• 3D

• Low temporal resolution• Spatial distributed• 1D (Line of Sight)

Goal: to take advantage of their complementary nature

Ground deformation monitoring

+ =Ground Deformation Map

Over the Whole Intestigated Area

The concept of the integration of GPS and DInSARdata to obtain 3D ground deformation maps

4.74.8

4.95

5.15.2

x 105

4.14

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x 106

0

2000

4000

6000

8000

GPS data

LOS ascending

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

450 0

0.01

0.02

0.03

0.04

0.05

0.06

=+DInSAR data

GPS measurements

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5.2

x 105

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0

500

1000

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3D Displacement vectors

3D ground deformation map

Two different methodologies have been proposed by:

Gudmundsson et al. (2002), Three-dimensional surface motion maps estimated from combinedInterferometric synthetic aperture radar and GPS dataJournal of Geophysical Reseach

Samsonov, Tiampo et al. (2006), Analytical Optimization of a DInSAR and GPS Dataset for Derivation of Three-Dimensional Surface MotionIEEE Geoscience and Remote Sensing Letters

Guglielmino, Nunnari, Puglisi, Spata (2009) submitted to IEEE.

GPS data

DInSAR Data

3D SurfaceMotion Map

Samsonov, Tiampo et al. (2006)

GPS data

KrigingInterpolation

DInSAR Data

Analytical Optimization 3D SurfaceMotion Map

Gudmundsson et al. (2002)

GPS data

KrigingInterpolation

DInSAR Data

3D SurfaceMotion Map

InterpolatedGPS data

Random Markow Field TheorySimulated Annealing Optimization

InterpolatedGPS data

Weighted Least Squares Method

Kriging Interpolation and Variogram model

Every components needs an appropriately choosen variogram model

The method we propose does not require a preventive interpolation of the GPS points, usually performed through kriging algorithm, thus avoiding the choice of a theoretical semivariogram, which is one of the main critical points in geostatistics.This choice, which is usually performed by supervising a preliminary statistic analysis of the experimental data, strongly affects the final result.

Small Deformation Theory

Let xo(x10, x20, x30) the position of an arbitrary point P surrounded by N points whose positionand displacements are respectively x(n)=(x1(n), x2(n), x3(n)) and u(n)=(u1(n),u2(n), u3(n)) .In a linear approach the small motions around a point P can be modelled by the N equations:

)3..1,()()( =+Δ= jiUxHu injijni

?

P1

P4

P3

P2

P5

P6

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=⊗+==

332313

232212

131211

)(21

εεεεεεεεε

ε jijiijij eeHHE

Strain tensor

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=⊗−==Ω

00

0)(

21

12

13

23

ωωωωωω

ω jijiijij eeHH

Rigid body rotation tensor

ijijji

ijij E

uu

H Ω+=∂

∂=

Displacement gradient

0)()( jnjnj xxx −=ΔRelative position

euAl +=

In a compact form the system of equation

)3..1,()()( =+Δ= jiUxHu injijnican be written as:

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔ−ΔΔΔΔΔ−ΔΔΔ

ΔΔΔΔΔ

ΔΔ−ΔΔΔΔΔ−ΔΔΔ

ΔΔΔΔΔ

=

)(2)(1)(3)(2)(1

)(3)(1)(3)(2)(1

)(3)(2)(3)(2)(1

)1(2)1(1)1(3)1(2)1(1

)1(3)1(1)1(3)1(2)1(1

)1(3)1(2)1(3)1(2)1(1

00001000000010

0000001............

00001000000010

0000001

NNNNN

NNNNN

NNNNN

xxxxxxxxxx

xxxxx

xxxxxxxxxx

xxxxx

A

}

}

First GPS point

Last GPS Point

Design Matrix

TUUUl ][ 231312232322131211321 ωωωεεεεεε=

Unknown parameters

TNNN uuuuuuuuuu ]...[ )(3)(2)(1)2(3)2(2)2(1)1(3)1(2)1(1=

Observation vectors

Small Deformation Theory

Weighted Least Squares approach

Weigh Matrix

W is the inverse of the data covariance matrix

WuAWAAl TT 1)( −=

euAl +=

In a compact form the system of equation

)3..1,()()( =+Δ= jiUxHu injijnican be written as:

A DInSAR interferogram can be related to the components U1, U2 and U3 of the displacement vector of an arbitrary point P according to the following equation:

where DLOS is the LOS displacements, at the point P on the Earth’s surface and V=[Sx Sy Sz]is a unit vector pointing from the point P toward the satellite.

Can we enter the DInSAR data in the small deformation system equations?

321 USUSUSD Pz

Py

Px

PLOS ++=

]000000000[ Pz

Py

Px SSSS =

lSD pLOS ⋅=

TUUUl ][ 231312232322131211321 ωωωεεεεεε=

tilosNNN Duuuuuuuuuu ]...[ )(3)(2)(1)2(3)2(2)2(1)1(3)1(2)1(1=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ−ΔΔΔΔΔΔ−ΔΔΔΔ−ΔΔΔΔ

Δ−ΔΔΔΔΔΔ−ΔΔΔΔ−ΔΔΔΔ

=

0000000000000100

00000100000001

............

............

............0000100

00000100000001

)(1)(2)(3)(2)(1

)(1)(3)(3)(2)(1

)(2)(3)(3)(2)(1

)1(1)1(2)1(3)1(2)1(1

)1(1)1(3)1(3)1(2)1(1

)1(2)1(3)1(3)1(2)1(1

Pz

Py

Px

NNNNN

NNNNN

NNNNN

SSSxxxxx

xxxxxxxxxx

xxxxxxxxxxxxxxx

A

321 USUSUSD Pz

Py

Px

pLOS ++=SISTEM Approach

Simultaneous and Integrated Strain Tensor Estimation from Geodetic and satellite deformation Measurements

WuAWAAl TT 1)( −=euAl +=

We emphasize that the SISTEM method is a point-wise oriented approach. This means that, at the unknown point P, SISTEM solves the WLS problem by taking into account the surrounding GPS points and only the DInSAR data coincident with the point P. Therefore, the spatial correlation of DInSAR data is not taken into account.

The variance of DInSAR data points was estimated directly from the interferogramby using a sample semi-variogram γ(hc) (Chiles and Delfiner 1999, Sudhaus and Jonsson, 2008

[ ]2

1)()(

21)( ∑

=

−=N

iiic sdrd

Nhγ

where hc is a classified separation distance

Scaling Function

)/exp()0/( 0ddddf −=

)/( oijI

ij ddfWW =

d0 is the level of localityof the estimation

Shen, 1996, Crustal deformation across and beyond the Los Angeles basin from geodetic measurements, Journal of Geophysical Research

Locality Effects• Only the point closer than about d0 to P give a significant contribution to the strain

estimate on P • The uniform distribution of the strain is required only in a neighborhood of each

computation point• For points P far away from GPS measumerent the DInSAR data becomes the dominant

information source

According to the modified least squares (MLS) approach proposed by Shen et al. (1996), based on the adjustment of the matrix W, we use the matrix WI which is a weighted version of the matrix W. Following the suggestion of literature [Teza et al., 2007, Shen et al. 1996] the weight function considered here is:

In this method, the only parameter that needs to be appropriately chosen is the parameter d0 in order to define the level of locality of the estimation. As suggested by Pesci and Teza (2007) we have related d0 with the mean inter-distance between neighbour stations. In particular let N be the number of EPs point of the network and Ki be the set of M nearest stations to the i station. We propose the following empirical formula to evaluate d0:

The optimal value of M depends on the topology of the network; based on several trials, we have empirically found that for random configurations M ranges between 4 and 6.

∑∑= ∈

=N

i Kjij

i

dNM

d1

01

P1

P4

P3

P2

P5

P6

Application of SISTEM: A synthetic case study

( ) ( )( )wyxezyxz /0

22

, +−=

Synthetic topography

Point pressure source (Mogi source)

2/322

3

)(43

dfdPax+

=Δμ 2/322

3

)(43

dffPaz

+=Δ

μ

μ=30 GPaf=5000ma3P=1017 Pa*m3

(a), (b), (c),(d) : East, North, Up and LOS components of the displacement field generated on the synthetic topography using the Mogi source

(e), (f), (g),(h): the three displacements components and the projected LOS calculated by the proposed GPS InSARintegration method (i), (l), (m),(n): residuals of the east, north and up component respectively. (o), (p), (q),(r): normalized histograms of the corresponding residuals errors, the mean value and standard deviation. A huge number of experiments performed allows to point out that the error distribution depends on the spatial distribution of the GPS point. In particular it was found that the best performance are obtained when a regular grid of GPS point is considered. Instead, if a randomly generated distribution of GPS point is considered the error distribution may result biased.

Error as a function of the number of EP

How it was expected accuracy increase with larger number of GPS points. However it can be appreciated that there are not sensible advantages in using a number of GPS point greater than 50-60 (see the plateau). Furthermore the best accuracy is achieved for the vertical component; this was an expected results which can be explained bearing in mind that the DInSAR images have an average vertical directional cosine of about 0.90 and therefore is particular sensitive to vertical movements.

Locality parameter considered

(a) dilatation; (b) differential rotation magnitude; (c) maximum shear strain

Application of SISTEM: the Mt. Etna 2003-2004 case study

This GPS dataset shows a significant inflation affecting the western and upper flanks, with a maximum of about 5 cm located on the upper southern flank, coupled with an eastward movement of the benchmarks located on the eastern flank of the volcano

An appropriate pair of ascending ERS2 SAR images was selected; they refer to the 20 August 2003 to 30 June 2004 interval and have a 70 m of perpendicular baseline.Interferogram was processed using the Jet Propulsion Laboratories (JPLs)/Caltech Repeat Orbit Interferometry Package (ROI_PAC, version 3.0).

The East component map show an evident displacement of the eastern flank. The most evident vertical movement (uplift) is localized in the summit western area according to a recharging phase of the plumbing system of the volcano during the investigated period [Bonaccorso et al. 2006].

The error maps of the horizontal components have similar patterns and show smallest error (blu area) where dense GPS coverage is achieved. The errors of the vertical component have a lower magnitude with respect to horizontal errors

10 15 20 25 30 35 40 45 50 555

10

15

20

25

30

N° GPS points

RM

SE

EastNorthUp

0 20 40 60-40

-20

0

20

40

60

80

100

120

GPS stations

Eas

t (m

m)

0 20 40 60-80

-60

-40

-20

0

20

40

60

GPS stations

Nor

th

0 20 40 60-100

-80

-60

-40

-20

0

20

40

60

GPS stations

Up

SISTEMGPS stations

SISTEMGPS stations

SISTEMGPS stations

RMSE (in mm) between the SISTEM output and the GPS data calculated at the GPS site locations as a function of the number of GPS stations

Discrepancy relevant to the East, North and Vertical components respectively, computed for the whole network configuration (i.e. 52 GP stations).

SISTEM Applications

520,000515,000510,000505,000500,000495,000490,000485,000480,000

4,196,000

4,194,000

4,192,000

4,190,000

4,188,000

4,186,000

4,184,000

4,182,000

4,180,000

4,178,000

4,176,000

4,174,000

4,172,000

4,170,000

4,168,000

4,166,000

4,164,000

4,162,000

4,160,000

4,158,000

4,156,000

4,154,000

4,152,000

3D clustering analysis performed by the Kohonen maps. This analysis was aimed to partition the whole displacement field into subsets sharing some common displacements features in order to recognize and classify deformation patterns affecting different sectors of Etna volcano

Abruzzo Earthquake case study:GPS, ALOS (ascending), ENVISAT(ascending and

descending)

+ + +

Conclusions

GPS and DInSAR integration based on small deformation theory

The proposed method was applied on the Mt. Etna area where the GPS network well cover the area and frequent SAR passes are available.

• GPS and DInSAR data are simultaneous integrated without the preliminary step of the Kriging interpolation

• Deformation field and relevant standard errors • Strain tensor and relevant standard errors • Rigid body rotation tensor and relevant standard errors

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