Automated procedure for InSAR data inversion using
Finite Element Method
Currenti G.1, Del Negro C. 1, Scandura D. 1, Williams C.A. 2
1) Istituto Nazionale di Geofisica e Vulcanologia – Sezione Catania
2) GNS Science – New Zealand
USEReST, Naples, 11-13 November 2008
Deformation model: Fault slip distribution
Dislocation source parameters determined from multiple types of static deformation data
such as GPS displacements, InSAR imagery, and surface offset measurements suggest
that slip along a fault is not uniform and is best described as a distribution of dislocation
sources.
di   Gij s j
j
displacement Green’s function slip
d  Gs
The procedure requires:
1) subdividing the faults in a finite
number of patches;
2) computing the Green's Function for
all the patches and the measurement
points;
3) solving a linear inversion problem to
determine the slip distribution.
At each node of the mesh different
values of Young modulus and Poisson
ratio are assigned on the basis of
seismic tomography investigations
(Patanè et al., 2006).
FEM generated Green’s function
Medium Heterogeneity
Young modulus
E  5 / 6Vp2
  [(Vp / Vs )  2] /[2(Vp / Vs )  2]
2
2
11.5-133 GPa
Poisson ratio
0.12-0.32
Real Topography
LaGrit
PyLith
Mesh
refinement
and
smoothing are available to
provide a mesh with more
resolution in areas of interest.
PyLith is a finite element
parallel code for the solution
of dynamic and quasi-static
deformation problems.
www.meshing.lanl.gov
www.geodynamcis.org
FEM Inversion
System of linear equations
d1  G11s1  G1M sM
d 2  G12 s1  G2 M sM

n x m patches
3 x n x m unknown
variables!


d N  GN 1s1  GNM sM
j≥3xnxm
(dip-slip, strike-slip, tensile)
Non-uniquiness of the solution can arise from limited data and poor knowledge of
the internal structure. This issue can be faced using: (i) as much data as possible
and (ii) numerical technique to reduce the algebraic ambiguity.
Geodetic Data Integration
 d1S   G1,1
   


 
 d N S   GN S , M
 1  
 U x   GN S 1,1
 U 1   GN  2,1
S
 1y   
 U z   GN S 3,1
   

 N  
U x G  GN S 3 N G  2,1
U N G   G
 y   N S 3 N G 1,1
U zN G   GN S 3 N G ,1










EDM, GPS, leveling data assure high
measurements
accuracy but the
coverage area is usually limited. InSAR
data, despite the lower accuracy, can
provide a better overview of the
deformation pattern thanks to the wide
coverage.




GN S , M 

GN S 1, M 
s 
GN S  2, M   1 
 
G N S  3, M   
  sM 


GN S 3 N G  2, M 
GN S 3 N G 1, M 

GN S 3 N G , M 
G1, M
Linear least-squares methods for this
problem
require
to
incorporate
regularization techniques in order to
stabilize the problem and to reduce the
set of likely solution.
d SAR  G SAR 
d   G  s
 GPS   GPS 
A Wd covariance matrix is introduced
to weight the data depending on
measurements uncertainties.


1

mind  min (Gs  d)T Wd T Wd (Gs  d) 
2

min  mind   r 
FEM Inversion & Regularization
The inverse problem can be re-formulated as an optimization problem aimed at finding
the unknown slip values s that minimize a data misfit and a smoothing functional:


1

min  min (Gs  d)T Wd T Wd (Gs  d)  sT WT Ws , L  s  U
2

As smoothing functional, the Laplacian Operator was used to avoid large variations
between neighboring dislocations.
The minimization of the quadratic functional φ subjected to bound constraints can be
solved by using a Quadratic Programming algorithm based on an active set strategy
(Gill et al., 1991):


1 T

min  min s Q s  bT s , L  s  U
2

where:
Q  GT WdT Wd G  WT W
b  GT Wd Wd d
T
L-curve criterion
The L-curves of the data misfit versus the model norm as a
function of the regularization parameter. The best value of
regularization parameter lies on the corner of the L-curve.
Input Fault Slip Distribution
φd
φw
Procedure Scheme
Ground deformation
data (SAR, GPS, EDM)
1 – Domain and Fault
discretization: LaGrit
3 – Solving a
linear inversion
problem: QP
algorithm
2 – FEM generated
Green’s function:
PyLith
Analytical vs Numerical Solution
We considered a 3D FEM model reproducing a rectangular dislocation source in a
homogeneous and isotropic half-space and compare it with the Okada’s solution.
A misfit index was used
to
quantify
the
discrepancy between
the numerical and the
analytical solution:
N
 i (U )   U
i 1
FE
i
U
N
AN
i
/  U iAN
i 1
Simulat.
#1
#2
#3
#4
#5
#6
#7
#8
Nodes
2228
2513
2570
3195
3977
5101
8591
36196
Elements
9858
11236
11505
14663
18888
25141
45021
205700
Quality
0.3544
0.3083
0.3441
0.3109
0.3417
0.2960
0.3050
0.2454
Currenti et al. PEPI 2008
Tests on the Accuracy
We compare the ground deformation
achieved
as
the
sum
of
the
displacements generated by each patch
and those obtained by the overall fault. In
these computations a uniform slip
distribution on the patch is assumed. The
error has a RMSE (root mean square
error) of the order of 10-7 m, which is well
below the measurement error.
The accuracy of the solution is
dependent on the number of the
observation
points.
A
uniform
distribution of observation points within
a regular grid was assumed, centered
on the ground projection of the
sources. The RMSE between the
assumed and the inverted slip is
computed
as
the
number
of
measurement points increases.
A case study: Northeastern flank movement at Etna volcano in 2002
On 22 September 2002, 1 month before the beginning of the flank eruption on the NE
Rift, an M-3.7 earthquake struck the northeastern part of Mt. Etna, on the
westernmost part of the Pernicana fault.
The GPS surveys carried out
in September and July 2002
shows a ground deformation
pattern
that
affects
the
northeastern
flank
clearly
shaped by the Pernicana fault.
Differential interferogram for ascending scene pair 31
July 2002 to 09 October 2002. The scale indicates the
phase variation along the LOS (negative values
correspond to the approaching of the surface to the
sensor)
Bonforte et al., Bull Volc. 2007
Source from GPS data Inversion
A numerical model was
set
up
using
the
dislocation sources from
GPS
data
inversion.
Topography and medium
heterogeneity are also
taken into account.
Bonforte et al., Bull Volc. 2007
3D FEM of Mt. Etna: slip on Northeastern Flank
Mesh Domain:100 x 100 x 50 km
Nodes: 129253
Tetrahedra Elements: 744886
Computing Time: 15 minutes
Sar Data and Source Discretization
Line of sight change map calculated by the unwrapped
interferograms, processed using ROI_PAC software,
developed by Jet Propulsion Laboratory & Caltech;
Courtesy of F. Guglielmino ed A. Spata.
A deformation pattern is clearly
seen around the Pernicana
volcano
edifice structure.
The
dislocation
sources were
subdivided in patches to obtain
the slip distribution from the
inversion procedure.
Green Functions Computations
248
fault patches
248 x 3 = 744
FEM simulations
FEM simulations
1
2
87
744
Dip-slip, strike-slip and tensile
dislocation were assigned to
the nodes lying along the fault
surface. For each patches, in
which the fault is discretized,
Green’s
functions
were
computed using PyLith. For
each each simulations, the
accuracy
was
warranted
checking the convergency of
the FEM solution. The iteration
of GMRES solver is stopped
when a threshold of 10-9 is
reached or the number of
iterations is higher than 200.
By a linear speedup on a cluster of
20 nodes the computing time
reduces from 10 days to 10 hours.
Inversion Results
19
18.5
Data Misfit
18
17.5
17
16.5
16
1000
0
1
2
3
4
Smoothing Functional
Inverted Solution
5
6
SAR data
7
-9
x 10
The kinematic of the faults was
constrained to those derived from
GPS data inversion. The solution in
correspondence of the corner of the
L-curve provides a deformation
pattern which resembles the SAR
data and also matches the GPS
obtained from the analytical solution.
Slip Distribution
Strike-slip, dip-slip and
tensile
dislocation
distribution.
Model Comparison
SAR data
Analytical Model
The GPS data inversion provides a
deformation pattern which misses the
clear anomaly around the Pernicana
fault
system.
A
heterogeneous
distribution of the slip along the structure
is able to better justify the SAR data. The
FEM model based on the numerical
inversion provides a more complex
pattern which is likely to be expected.
Numerical Model
Conclusions
An automated procedure for geodetic data
inversion is proposed to estimate slip distribution
along fault interfaces. 3D finite element models
(FEMs) are implemented to compute synthetic
Green’s functions for static displacement.
FEM-generated Green’s functions computed
using PyLith, a source-free parallel finite element
code, are combined with a Quadratic
Programming algorithm to invert ground
deformation data and obtain an estimate of slip
distributions along seismogenic structures.
The procedure was applied to study the ground deformation preceding the 2002-03 Etna eruption.
SAR images showed a significant deformation pattern of the north-eastern flank of the volcano
involving the main local volcanic edifice features on the northeastern flank. The numerical model
highlighted a heterogeneity slip distribution along the Pernicana fault with a predominant strike-slip
mechanism associated with a dip-slip movement in the western part.
The time consuming computation of the procedure is related to the generation of the Green’s
functions. For the main volcano edifice features and the seismogenic structures they has to be
computed only once and stored in a database.
Numerical Solution
Several computations were
performed
to
better
understand
the
effects
induced by the topography
and
the
medium
heterogeneity.
The
topography
significantly
alters the general pattern of
the
ground
deformation
especially near the volcano
summit. On the contrary, the
medium heterogeneity does
not strongly affect the
expected deformation.
Currenti et al. PEPI 2008
Smoothing Functional




1 1 T

T

T
TT
TT
min min
(Gs (dG) sW
(
G
s

d
)


(
s

s
)
W
W
(
s

s
)
 dd )TW
W
W
(
G
s

d
)


s
W
W
s

dd
0
0 
d
LL
LL
2 2


The Laplacian Operator is introduced to avoid large
variations between neighboring dislocations
 s0
2
si 1, j  2si , j  si 1, j
(x)
2

si , j 1  2si , j  si , j 1
(y) 2

The matrix WL is constructed so that
the n-th row in WL contains
coefficients in the equation above for
columns corresponding to the
neighboring source.
W( j 1) nc i  2 x 2  y 2

2


W


x
 ( j 1) nc (i 1)

WL  W( j 1) nc (i 1)  x 2
2
W




y
 ( j 2) nc i
2
W jnci  y 
