On the determination of the space-time slip distribution via Linear Programming
Techniques
B. Caldeira
(1)
, V. Bushenkov (2) , G. Smirnov (3) , J. Borges
(1)
, and M. Bezzeghoud
(1)
(1) CGE, Department of Physics, University of Evora, Evora, Portugal
(2) CIMA, Department of Mathematics, University of Evora, Evora, Portugal
(3) Centre of Physics, Department of Mathematics and Applications, University of
Minho, Braga, Portugal
Reference author:
Bento Caldeira
Geophysical Centre of Évora and Physics Department, ECT, University of Évora
Colégio Luís António Verney
Rua Romão Ramalho, 59
7002-554 Évora
Tel.: +351 266 740 800
E-mail: bafcc@uevora.pt
1
Abstract
The purpose of the seismic source investigation is to obtain an accurate description of
the rupture from seismic and geodesic data. The success of this depends fundamentally
on three factors: source models, informative content of the data, and the methods used.
Currently, inversion techniques are considered better methods for obtaining rupture
characteristics from seismic and geodesic data. In this paper we suggest to formulate a
Linear Programming (LP) inversion in DUAL formulation, for reconstructing the
kinematics of the rupture of large earthquakes through space-time seismic slip
distribution on the faults plane. In its general setting, the algorithm can produce results
from strong ground motion waveforms, but can also be used with teleseismic
waveforms as well as with geodesic data (static deformation). To test the algorithm and
examine its stability, and robustness we define a synthetic rupture model, based on a
real space-time slip distribution scheme. We compare the proposed approach with
others standard algorithms, rehearsing reconstructions with same synthetic waveforms
through these algorithms. Green functions were calculated by a finite differences
method with a 3D structure model.
1 - Introduction
The description that one can have of the seismic source is the manifestation of an
imagined model, obviously outlined from Physic Theories and supported by
mathematical methods. In that context, the modelling of earthquake rupture consists in
finding values of the parameters of the selected physics-mathematical model, through
which it becomes possible to reproduce numerically the records of earthquake effects
on the Earth’s surface. Actually, these effects are the elastic records at near field
source, elastic records at far field source, and inelastic deformations recorded by
geodetic techniques. The detail and accuracy level, with which the characteristic
parameters for large earthquakes are computed, depends on the combination of two
factor classes involved in the process - used methods and data.
The kinematic model of finite seismic source consists in a spatiotemporal distribution
of slip vectors on a fault plane, previously divided into a grid of sub-faults. The entire
parameters set that characterize the vector slip of each sub-fault is as follows: init time,
amplitude, direction and temporal evolution (or, alternatively, rise time).
Currently, the most popular approaches to determine the slip distribution models are
the inversion of near-source ground motion data (e.g. Asano and Iwata 2009; Suzuki et
2
al., 2009; Hartzel et al., 2007) and the joint inversion of near source and teleseismic
waveforms (e.g. Delouis et al., 2009; Yagi, 2004; Mozziconacci et al., 2009). The
reason to prefer near source data is that it allows to reconstitute the rupture kinematics
with more details than when teleseismic waveforms are only used. The main
difficulties when trying to use this kind of data are as follows: at first, the lack of
accelerometer coverage in some relevant seismic zones, at second, models of Earth
structures must be accurate, that don’t occurs for many zones, and, at last, very high
computation power is required for such waveforms modelling.
According to Tarantola (1987), the study method of any physical system, whether it is
a quantum particle, a galaxy or the Earth, is developed following the methodological
proposal of the Discrete Inverse Theory that involves
forward problem,
parameterization and inverse problem. In the problem discussed in this paper, the
forward problem is implemented by the operational module that computes the soil
movements
(elastic displacement, velocity, acceleration or deformation) for all
observation points (stations). The central trait of the forward problem is the calculus of
the Green’s functions. They consist to find an approximate solution of the secondorder elastodynamic equation for a point in an elastic medium when perturbed by a
point unitary source using the known structure model and source mechanism model.
There exist some approaches to resolve the problem. By using 1D layered Earth
structure models it is possible to find reasonable Green’s functions at low frequencies
(<1Hz) applying such methods as Kennett and Kerry, 1979; Bouchon, 1981; Bernard
and Madariaga, 1984; Spudich and Fraser, 1984; Sikia, 1994. The Green’s functions
describing wave propagation through complex 3D anisotropic structures can be
calculated by algorithms based on finite differences (e.g. Olsen and Archuleta, 1996;
Pitarka et al., 1998; Olsen, 2000; Larsen & Schultz, 1995), finite element (e.g. Bao et
al., 1998), or also spectral elements (Komatitsch 1997, Komatitsch & Villote, 1998;
Komatitsch et al., 2002, etc).
The first attempts to invert waveform to characterize spatial and temporal rupture
processes of finite seismic sources were made in theoretical studies (see Gilbert, 1975;
Hartzell et al. 1978) or applying to real situations (for example, San Fernando 1971
earthquake - Trifunac, 1974; Langston, 1978; Heaton, 1982; or the Imperial Valley
1979 earthquake - Hertzell and Helmberger, 1982). These early works represent the
rupture model in the form of a succession of slips at sections of the rectangular fault
plane. The init time of each section (sub-fault) was commanded by a rupture front that
3
spreads over the fault plane with constant velocity in all directions from the hypocentre.
The evolution of each slip was given by a temporal function of certain shape. This
source model scheme is known as model of simple time window. There were two main
aspects difficult to resolve in the original models of single time window. The first is
related with the shape and duration of the source time functions that is equal for all
subfaults; that limits the frequency range of the data modelled. The second is the
improper assumption to impose a rupture with constant velocity, which also affects a
correct data modelling. These two difficulties were partially resolved by the model of
multiple time windows (eg. Olson and Apsel, 1982; Cohee and Beroza, 1994). In the
multiple time windows model, the evolution of slip in each sub-fault is represented by
a succession of elementary source time function, which imposes that the rupture of
each sub-fault occurs in separate time intervals. The multiple time windows version
ensures a more realistic simulation of the ruptures although with significant increase of
calculation efforts.
In the essential, the present framework of finite-source models (eg. Ammon et al, 2005,
Asano and Iwata, 2009; Mozziconacci et al., 2009, Robinson and Cheung, 2010;
Delouis et al., 2009) are not much different from previous works; the major
transformations refer to the grown of the computation scale and techniques imported
from optimization.
The search of the source parameterization that ensures the best fitting between
observed and synthetic waveforms constitutes an inverse problem, whose can be
solved by different methods. When the problem is posed to estimate whole source
model parameters to large earthquakes, constitutes a nonlinear configuration.
Otherwise, if it is possible to connect synthetic data with a model of linear equations,
the problem can solve using the techniques of linear inverse theory (eg. Tarantola,
1987, Menke, 1984). The local inversion methods, as least squares, when applied to
linear systems of large dimensions as seismic waveform inversion, are unstable due the
existence of many local minima. In these cases is requires the introduction of
stabilization factors physically reasonable: the constraints. The positivity that prohibits
negative seismic moment values, is a constraint naturally assumed when used the Non
Negative Least Squares algorithm (NNLS) (Lawson and Hanson, 1974) to inverts
seismic waveforms to slip distribution (eg. Hartzell and Heaton, 1983; Ide et al., 1996;
Suzuki et al., 2009). Minimum norm, minimum roughness, fix the total moment and
fix the rupture velocity are also constrains used to regularize the inverse problem.
4
When the formulation of the nonlinear problem is applied to earthquake source
inversion, there are some techniques to solve it. Global search algorithms have been
implemented to explore all solution domains. Two of these algorithms extensively used
are the Simulated Annealing (SA), and the Genetic Algorithm (GA), both developed
from inspiration from nature's processes that try mimicking. The SA algorithm uses the
sophisticated Monte Carlo random method (Kirkpatrick et al., 1983) to simulate the
annealing in thermodynamics systems. The GA (Holland, 1975) operates on analogy
with the evolution of biological populations. Ihmlé (1998) uses SA to make inversions
of seismic data for the distribution of slip; Hernandez (2001) obtains slip distributions
applying GA. Another variant of these methods with ability to be used in nonlinear
inversions is the Neighbourhood algorithm (Sambridge, 1999). The inversions of
seismic data to kinematic finite-fault slip distributions can adopt either objective
functions of L1-norm, as L2-norm. Das and Suhadolc (1996) and Hartzel et al. (2007)
analyse the important differences existed between L1 and L2 norm inversions.
In this paper we present and test a Linear Programming (LP) inversion in Dual form,
for reconstructing the kinematics of the rupture of large earthquakes through spacetime seismic slip distribution on finite faults planes. The proposed method can be
considered as a continuation of the work started in Das and Kostrov (1990). The
proposed algorithm uses strong ground motion waveforms, but it can also used with
other types of data as teleseismic waveforms as well as with geodesic data (static
deformation). To test the method a synthetic model was defined to compute
seismograms that were inverted using the same approach as for the real data. We
compare the proposed approach with others standard algorithms, rehearsing
reconstructions with same synthetic waveforms through these algorithms. Green
functions were calculated by the finite differences method applied to a 3D structure
model.
2 - The forward problem
The description of the elastic displacement produced at the Earth surface, as
consequence of applied body forces or slip discontinuities in a semi-infinite elastic
medium, constitutes the fundamentals for formal development of the methods to
seismic source study. The representation theorem in its integral form (e.g. Aki and
Richards, 1980; Ben-Menahem and Singh, 1981; Udias, 1999) is the starting point for
5
the construction of physical-mathematical formalism that supports the kinematic
methods:
+∞
𝜕
𝑢𝑖 (𝑥𝑗 , 𝑡) = ∫−∞ 𝑑𝜏 ∫Σ 𝐷𝑚 (𝜉, 𝜏)𝐶𝑚𝑛𝑜𝑝 𝜈𝑛 𝜕𝜉 𝐺𝑚𝑖 (𝑥, 𝑡 − 𝜏, 𝜉)𝑑Σ
(1)
𝑝
where 𝑢𝑖 (𝑥𝑗 , 𝑡) represents the ith component of the seismic displacement at observation
point 𝑥𝑗 , and time t; 𝐷𝑚 (𝜉, 𝜏) is the mth component of the slip produced at time
the position
on
𝐶𝑚𝑛𝑜𝑝 is a tensor that depends of the elastic
proprieties and the geometry of the fault plane; 𝜈𝑛 is a vector perpendicular to the fault
plane; and the 2nd order tensor 𝐺𝑚𝑖 (𝑥, 𝑡 − 𝜏, 𝜉) is the Green Function, that represents
the temporal evolution of the i component of the displacement at the position 𝑥𝑗 , due to
a unitary slip in m direction produced at source, in position
and time
.
The discretization of the integral form of the representation theorem (1) through proper
parameterization of the source is the tool that allows us to compute the synthetic
seismograms. To do this, the fault plane of the finite-source is discretized into a set of
N sub-faults defined by a grid covering the entire surface and disposed along a square
orthogonal referential xOy, where Ox is the strike-axis and Oy is the dip-axis. Also the
total time of rupture needs to be discretized into Nt steps of time. Each subfault l
(l=1,N) constitutes a point source that at certain time step k (k=1,Nt) initiates slips
̇
(breaks), according to a source time function 𝑆𝑘,𝑙,𝑚
(𝑡) in direction m. The slip vector
is defined through the magnitude of the two orthogonal components m; one in the
strike direction (m=1) and the other in the dip direction (m=2).
The rupture described by this model is a sequence of slips, each one characterized by
own: a) position, b) initial time, c) amplitude, d) direction, and e) source time function.
The adopted finite-source model allows yet that each subfault after break once, can be
reactivated and break later, in different stages of the rupture.
The complete
parameterization of this model requires yet defining the geometry of the fault plane;
hypocenter position; size of each sub-fault; as well as the interval of the time-step in
which rupture was discretized.
𝑗
The ith component of displacement at station j, 𝑢𝑖 (𝑡), is calculated by
𝑗
𝑁
𝑡
2
̇
∑𝑁
𝑢𝑖 = ∑𝑘=1
𝑙=1 ∑𝑚=1(𝑆𝑘,𝑙,𝑚 (𝑡) ∗ 𝐺𝑖,𝑗,𝑘,𝑙,𝑚 (𝑡)) 𝑥𝑘,𝑗,𝑚
(2)
where the indices i,j,k,l,m represent: i – direction of the displacement at observation
point (1=X, 2=Y, 3=Z); j – observation point; k – time step were the time of rupture
was discretized; l – sub-faults; m – components of the slip vector (1 – strike direction,
6
2 – dip direction); the asterisk (*) denotes the convolution and 𝑥𝑘,𝑙,𝑚 is the slip. We
̇
approximate the source time function 𝑆𝑘,𝑙,𝑚
(𝑡) within each cell along strike and dip by
a triangular function with unitary area.
The Green functions were calculated considering propagation of seismic waves in a 3D media. We used the code E3D, an explicit elastic finite-difference wave propagation
code (Larsen & Schultz 1995), based on the work of Madariaga (1976).
The system of linear equations (2) that allow compute the synthetic seismograms
(forward problem) can be translated to matrix language by the multiplication of
matrices,
u = A𝐱
(3)
where u is the vector that contain all the seismograms; x the vector of the slips of all
subfaults in whole time steps; and A matrix of numerical parameters to compute the
synthetic seismograms - the matrix of synthetics. The matrix A contains Green
Functions relating slips in each subfault at each time-step. The set of Green Functions
at different time steps differs only on the init time; can be obtained, to subsequent time
steps, merely by a shift in time (delay) applied in the first set of them. Fig. 1
summarises the formulation of the forward problem.
Under the hypothesis of fix slip direction and constant rise time of individual source
time functions, the problem of complete seismic space-time slip distribution
reconstruction reduces to the solution of a system of linear equations. It is well-known
that this inverse problem is ill-posed Error! Reference source not found.. The usual
regularization techniques Error! Reference source not found. can hardly be applied
in this case because of a very high dimension of this problem. The problem can be
overcome by introducing some additional regularizing constraints. Some additional
physical hypotheses, like no-backslip constraint, result in condition of nonnegativeness of solutions to the system of linear equations. This makes natural to
consider the inversion of the slip time history and distribution in the frame of linear
programming theory. The linear programming approach was first applied to solve this
problem in Error! Reference source not found.. This seems to be a unique
reasonable method since as it is well-known Error! Reference source not found. the
least-squares method is unable to solve the inversion problem due to the fact that many
negative values of moment rate, which did not exist in the forward problem, are
produced.
7
The hypothesis of constant slip direction in general is not verified and the "real"
seismic slip time history and distribution reconstruction becomes an hard nonlinear
problem. In this work we suggest an algorithm for seismic slip time history and
distribution reconstruction allowing to solve the problem in its general setting. The
solution of an auxiliary linear programming problem is an essential part of the
developed method and, from this point of view, our wor can be considered as a
continuation of the research started in Error! Reference source not found.. To test
the algorithm we use a synthetic displacement function for the fault model and perform
the inversion.
3 - The inversion algorithm
The slip determination problem can be formalized in the frame of mathematical
programming in the following way
c, x  min ,
A( ) x = b,
x  0.
Here
x
(3)
is
the
unknown
vector
of
amplitudes
and
residuals
(see
Error! Reference source not found.) and the vector  represents the unknown rakes.
Note that the displacement field models can be different but the mathematical
formalization is always the same. If we fix the rake vector  , problem
(Error! Reference source not found.) becomes a linear programming problem. This
observation
is
the
key
to
an
effective
solution
of
problem
(Error! Reference source not found.). It turns out that the gradient of the minimized
functional c, x with respect to  can be calculated in terms of the solution to the
linear programming problem dual to (Error! Reference source not found.).
The following algorithm describes the process.
Algorithm:
Given 0 ,  > 0 , and  > 0 .
for k = 0,1,2,
Step 1. Solve linear programming problem (Error! Reference source not found.)
with  = k and obtain xk .
Step 2. Obtain search direction  k and a step  k > 0 .
8
if  k Pk P <  break
else
Step 3. Set k 1 = k   k k .
end(for)
The second step of the algorithm is not trivial and we give its detailed description.
Choose any direction  and  > 0 . Since the matrix A is a smooth function of  , we
have A(   ) = A( )  A ( )  o( ) . Let x( ) be a solution to the following linear
programming problem:
c, x  min ,
( A( )  A ( )) x = b,
x  0.
(4)
We assume that the matrix A( ) has full row rank. Without loss of generality we have
the representations A( ) = [ B( ), D( )] and A ( ) = [ B ( ), D( )] , where B( ) is a
basis
matrix
(see
Error!
Reference
source
not
found.)
for
problem
(Error! Reference source not found.) with  = 0 . By c B we denote the basic
components of the vector c . If  is sufficiently small, the solution to problem
(Error! Reference source not found.) has the form x( ) = [ xB ( ),0] . Since
B( ) xB (0) = b and
( B( )  B ( ))( xB (0)  x B (0)  o( )) = b,
We get
x B (0) = ( B( )) 1 B ( ) x B (0).
Let y ( ) be a solution to the dual linear programming problem:
b, y  max ,
( A( ))T y  c.
(5)
Then we obtain
d
c, x( )
= cB , ( B( )) 1 B ( ) xB (0)
d
 =0


=  ( B( )) 1 cB , B ( ) xB (0) =  y( ), B ( ) xB (0).
T
This formula gives a possibility to calculate the gradient V ( ) of problem (5) value
V ( ) . Indeed, taking  equal to i th basis vector, ei , we get V ( )/i .
9
Thus, we obtain the search direction k = V (k ) and choose the step  k > 0 as the
maximal  ]0, ] satisfying the condition ( xk ) B   ( B(k )) 1
 B (e )( x )
i
i
k B
 0.
.
Let  > 0 . The function g Q () (elementary moment rate function) has the form shown
in Fig. 1.
5 - Initial point for the simplex method
Let x  R n . By ( x)i we denote the i -th coordinate of x . By e p we denote the vector
with
0, i  p,
(e p ) i = 
1, i = p.
Consider the problem
c, x  min ,
Ax  0,
 1  ( x) i  1, i = 1, n,
(1)
where A is an (m  n) -matrix with m < n . Suppose that rank A = m . Obviously x = 0
is an admissible point for problem ((1)). To find an initial point for the simplex-method
we use the following algorithm.
Set m0 = m , P0 =  , A0 = A , b0 = 0 , and x0 = 0 . Assume that we have already
constructed number mk , a set of indices Pk , an (mk  n) -matrix Ak , a vector bk  R
mk
,
and a point xk  R n , such that Axk = bk . If mk = n , then the point xk is a vertex of the
polyhedron X = {x  R n | Ax  0 | ( x)i | 1} .
If mk < n , we construct the next quintet (mk 1 , Pk 1 , Ak 1 , bk 1 , xk 1 ) in the following way.
Using the Gauss elimination algorithm we obtain a representation Ak = [ Bk , Ck ] , where
Bk is an upper triangular (mk  mk ) -matrix and Ck is a rectangular (mk  (n  mk )) -
10
matrix. Consider a non-zero vector zk  R
vector xk = ( yk , zk )  R
mk
R
n  mk
n  mk
and put yk =  Bk1Ck zk  R
mk
. The
satisfies Ak xk = 0 . Set
tk = sup{t || ( xk  txk )i |< 1, i 
 Pk }
and
xk 1 = xk  t k xk .
Obviously we have
Ak xk 1 = Ak xk  t k Ak xk = bk .
Put
Pk 1 = {i1 ,il } = {i = 1, n || ( xk 1 )i |= 1}.
k 1
Denote by mk 1 the sum of m and the cardinality of Pk 1 . Define the matrix Ak 1
adding to the matrix A the rows e p , p  Pk 1 . The first m components of the vector
bk 1 are equal to zero and (bk 1 )i = ( xk 1 )i , i  Pk 1 . Therefore Ak 1 xk 1 = bk 1 .
By induction, after at most n  m steps we get a vertex of X .
6 - Data to evaluation
To evaluate the proposed algorithm, we applied it to a synthetic scenario of seismic
rupture similar to real sources. This kind of evaluation is extremely important since it
represents the only reliable way to analyse the performance of the methods where the
expected results are known (Beresnev, 2003).
The synthetic waveforms used were calculated by Eq. 3 for a defined rupture model, to
a set of 13 seismic stations distributed around the source as the geometry represented
in Fig. 3. To each considered station were calculated the 3 components (N, E and Z)
velocity waveforms.
The seismic rupture was defined by a spatiotemporal distribution of slips dominated by
thrust with left-lateral strike-slip component on a rectangular fault plane of 10 x 10
km2, oriented towards the N and dipping to East (dip is equal to 45 deg.). The fault
plane was divided into a grid of 36 square subfaults, with size of 2 x 2 km2. Each
individual slip vector (placed at each node of the grid) is characterised by two
orthogonal components, in strike and dip directions,
of each source is
=atan(
2/
1).
1
and
2,
where the rake angle
The rupture starts at initial time at hypocentre node
at 12 km depth and spreads in all directions with a variable velocity. The slip of each
11
node is specified by the initial time, two components of the slip vector and a triangular
Source Time Function (STF) with rise time
. The rupture time is discretized using a
temporal gridding of 0.3s. The defined source model assumes that some subfaults slips
more than once at different stages of the rupture. Table 1 and Fig. 4 represents the
space time slip distribution used and the complete STF. The Green Functions were
computed using a finite-difference spatial and temporal grid spacing scheme of 0.5 km
and 0.03sec respectively. The velocity model considered is a fragment of
100kmx100kmx70km of the 3-D velocity model of SW Iberia (Grandin et al. 2007).
Given that the Green Functions simulated by the finite-difference method are only
reliable at a maximum frequency (fmax) propotional to the minimum velocity (vmin), and
the inverse of the grid spacing (h) given by fmax < vmin/5h (Pitarka, 1999), with the
parameterization chosen, and the used velocity model, we generating a maximum
frequency less of 1.4Hz. Thus the Green Functions calculated were filtered with a low
pass Butterworth Filter with a cut-off frequency of 1.3Hz.
7 - Inversion Results
From the same data were performed inversions using three different algorithms. Two
standard - the least-squares of Lawson and Hanson (1974) through the formulation of
Hartzell and Heaton (1983) and the Linear Programing by Primal formulation
developed by of Das and Kostrov (1990); and a Dual version of Linear Programming
presented in this paper. All the procedures have been used with same Green functions
and source parameterization. Due the phenomena of generation and propagation of
seismic waves, the amplitudes of the waveforms vary depending on the distance and
relative position from the source. To prevent data calculated/recorded with the same
accuracy may produce different weights in the objective function, the amplitude of
waveforms was normalized.
The reconstructed rupture model calculated using the Dual version of Linear
Programming algorithm developed (Fig. 5) is comparable with the synthetic base
model (Fig. 4). The likeness between the two models is evident in both the spatial
distribution of slip as in its spatial occupation over time, characterized by a nonuniform rupture front. The “Total” slip distribution of the synthetic and reconstructed
models (comparison of Figs. 4 and 5) proves the spatial likeness; the evolution of the
rupture, displayed in the sequence of snapshots and in STF of synthetic and
reconstructed models, proves the suitable temporal reproduction. The comparison
12
between the observed and model-predicted waveforms is in many real situations the
only way to validate the reconstructions. Sometimes the two seismograms are simply
displayed together to visual comparison; other times they are applied quantification
criteria to characterize differences (eg. Geller and Takeuchi, 1995; Kristekova et al.
2006). In this paper we use the he Normalized Root-Mean-Square Deviation
(NRMSD) method to compare seismograms,
∑𝑡(𝑢(𝑡) − 𝑢REF (𝑡))
√
𝑁𝑅𝑀𝑆𝐷 =
𝑁𝑡(𝑢max − 𝑢min )2
where 𝑢(𝑡) is the tested seismogram, 𝑢REF (𝑡) is the reference seismogram, Nt is the
number of elements of the seismogram, 𝑢max the maximum amplitude of the
seismogram and 𝑢min the minimum amplitude. Fig. 6 shows the synthetic and
reconstructed models, and respective predicted waveforms. The similarity of the two
models explains the good fit between the two sets of waveforms (NRMSD=1,23%).
The same parameterization was applied to inverts by Primal formulation of Linear
Programming. It was implemented using the same inversion code (reference to
algorithm) applied to the dual version. To this case the convergence was also obtained,
to the same solution, but after a processing time about 100 times higher than necessary
from Dual formulation (~12 min with DUAL and ~14H with PRIMAL) and with a
great number of warnings.
Inversions by the formulation of Hartzell and Heaton (1983) don’t estimate the starttime slip of sub-faults. Is a parameter fixed considering that exist a rupture front
spreading with constant velocity from the hypocenter, and that the start-time slip at
each sub-fault coincides with the arrives of the rupture front. It´s a constraint that
promotes a high reduction of the number of parameters to estimate and consequently
makes possible the utilization of local inversions as NNLS of Lawson and Hanson
(1974). This method was used with a rupture velocity fixed at 3km/s and the other
parameterizations were the same used in the formulations of linear programming. The
reconstructed model by this method (Fig. 7) differs significantly from the synthetic
origin model used (Fig. 4), both in total slip distribution as well as their distribution in
time. Surprisingly this divergence between models occurs despite the good fit on the
two sets of waveforms (NRMSD=4,73%, Fig. 8), where can prove that different
models can produce very similar waveforms.
13
8- Discussion and conclusions
The reconstructing of the kinematics of the rupture for large earthquakes from a set of
their effects recorded on the Earth surface constitutes a scientific problem, whose
solution consists to determinate the evolution of the slip vector over the fault area. The
inversion formulations to produce physically consistent solutions must be stated in its
global formulation, which evolves extensive models. The Linear Programming (LP)
techniques using the PRIMAL formulation (Das Kostrov, 1990; Hartzel and Liu, 1995)
revels an appropriate tool to make this since the scale of the problem does not reach
large dimensions. Is a inversion method that explores full space of solutions. The
easily to incorporate constraints that improves the convergence, since reduces the
number of vertices in the hyperspace of the solutions, is an advantage of this kind of
approaches relatively the iterative methods. However, if the problem of inversion is
parameterized in order to involve a large number of equations, its resolution through
the PRIMAL LP inversion techniques can become a problem difficult to solve with
conventional computation means. To these cases, that occur when we want to
reconstruct the kinematics of the rupture for great earthquakes, we suggest to formulate
a Linear Programming inversion in DUAL formulation. It reduces the dimension of the
variable space, put the observed data (u) in the objective function and consequently
makes the computation process more stable. We show that the DUAL formulation
presents clear advantages in both the convergence and computing time relatively to the
PRIMAL formulation used by other authors.
The ability that the slip inversion methods have to build detailed scenarios of rupture
makes them one of the more attractive tools to study the seismic source. However, a
detailed analysis of the solutions acquired by these methods, warns for a set of cautions
that must be present before use of these tools. The first is the choice of method.
Looking at the literature we found a number of different ways of obtain inversion
scenarios. All of them consider equivalent levels of respect to physical and numerical
requirements (eg Hartzel and Heaton, 1983, Hernandez et al. 2001; Valle and Bouchon,
2004). However, when apply to the same events, with same data, show different results,
as can be compared from the works of Wald and Heaton (1994) and Cohee and Beroza
(1994), for the Landers earthquake of 1992. Selected the method is necessary fix the
parameterization: the geometry of the fault; size for the sub-faults; shape of source
time functions; number of time windows in each subfault,... . Arise evidences that
14
different parameterizations lead to different results, but equally plausible. Because the
complex nature of the problem that does not allow knowing the best solution, is not
possible use real data to compare methods or investigate parameterization schemes,
based on analysis of solutions (Beresnev, 2003). These studies can only be performed
with synthetic data calculated from rupture models defined. Therefore we tested the
stability and robustness of the alghoritm using synthetic waveforms computed from a
defined slip distribution model.
The results reveal the good likeness between the reconstructed model, using the DUAL
LP inversion algorithm proposed and the known source model.
The reconstruction tested with the same synthetic waveforms, but using the NNLS
algorithm (considering the rupture propagation with constant velocity), differs from the
model used, although showing a good fit of data (Fig?).
We note that the two versions of LP inversion formulations (PRIMAL and DUAL)
converge in the same solution, however with very different computing times. Using the
same simplex inversion routine, the DUAL converge after about ~40min whereas the
PRIMAL after ~14h. The method presented in this paper can be generalized to be used
jointly with other data types (Geodetic and teleseismic waveforms).
The results obtained in this paper encourage us to apply the algorithm proposed with
real seismic and geodetic data.
Acknowledgments
This work has been developed with the support of the ‘Fundac¸a˜o para a Cieˆncia e a
Tecnologia (FCT)’ (Science and Technology Foundation) of the ‘Ministe´rio da
Cieˆncia, Tecnologia e do Ensino Superior (MCTES)’, through the project
SISMOD/LISMOT (PTDC/CTE-GIN/82704/2006).
15
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20
FIGURES
Fig. 1 – A) The 3 components (X,Y,Z) of synthetic seismograms to 2 stations (Station
1 and 2) produced by a slips in two directions from three subfaults (green, orange and
blue) on a space time discretized fault plane are computed B) by the matrix product
between Green Function’s of each subfault/station at each time-step, convolved with
source time function, and the slips of correspondent subtault in same time-step.
Fig. 2 – Geometric representation of DUAL formulation Solution
Fig 3 – Geometry of the situation tested. At the surface the 13 triangles represent the
stations; at the interior of the volume the fault plane.
Fig. 4 – Individual panels (1 to 5) show the slip amplitude cumulative distribution
(colorscale) at time windows of 0.78s. The panel (Total) shows the final slip
distribution (red arrows) and the colorized contours show rupture time in 0.6-sec
contours; The panel identified as “Complete STF” represents the rate of moment
libertação
Fig. 5- Comparison the original used source model (left) and the reconstructed by
inversion of DUAL simplex method (right). The top trace is the stack power spatially
integrated over the grid as a function of time, with red lines indicating the times of the
spatial snapshots. Snapshots from the continuous back-projection (see Supplementary
movie 1) indicating intervals in which the normal fault radiation dominates (during the
first 50 s, typified by the 25 s frame), and when secondary events on the megathrust or
in the upper plate occur (frames at 52 s, 91 s and 118 s). At 52 s, there is simultaneous
radiation from the normal fault and the megathrust. The event at 118 s is probably in
the upper plate, and although it is strong in the short-period radiation, it is not strong at
long periods. The full animations for F-Net and five other networks are shown together
in the Supplementary Information.
Fig. 6- At left - Comparison the seismograms produced from the original used source
model (blue) and reconstructed model (red); at right – total slip distribution of the two
models: up- the original; bottom- the reconstructed from simplex DUAL.
21
Fig. 7- Comparison the original used source model (left) and the reconstructed by the
NNLS method (right).
Fig. 8- Left - Comparison the seismograms produced from the original used source
model (blue) and reconstructed model by the NNLS method (red); right – total slip
distribution of the two models: up- the original; bottom- the reconstructed from NNLS.
Fig. 9- Comparison of the temporal evolution of the seismic moment rate (STF) of the
three models: the original (blue); the reconstructed by Linear Programming (red); the
reconstructed by NNLS (green).
22
Fig. 1
zj
1
u
-1
1
zi
Solution
space
-1
Fig. 2
23
Fig. 3
Fig. 4
24
Fig. 5
25
Fig. 6
26
Fig. 7
27
Fig. 8
Fig. 9
28
Tabeles
Table 1
Position Position Slip
in strike in
dip strike
in Slip
in
-dip
axis
axis
direction direction
(km)
(km)
0
0
1.70
1.41
0
-2
4.83
-2
0
2
Init
time
Position Position Slip
in strike in
dip strike
in Slip
in
-dip
time
axis
axis
(km)
(km)
0.0
-4
6
2.20
3.31
2.1
8.05
0.6
4
6
2.72
3.28
2.1
0.54
0.43
0.6
-6
0
0.11
0.07
2.4
0
3.19
2.93
0.6
-6
4
2.08
2.21
2.4
0
2
2.79
3.49
0.6
0
8
2.06
2.56
2.4
4
2
4.46
6.68
0.6
2
8
0.12
0.09
2.4
-4
0
4.83
6.38
1.2
4
8
2.68
3.75
2.4
4
0
0.11
0.14
1.2
-4
4
4.13
2.54
2.7
-2
4
3.16
3.97
1.2
-6
2
4.92
3.41
2.7
0
4
4.16
3.47
1.2
-4
8
0.32
0.22
2.7
-2
-2
1.48
2.08
1.5
-2
8
1.24
1.27
2.7
2
-2
3.11
4.48
1.5
-6
8
0.63
0.37
3.0
-2
2
0.89
1.26
1.5
2
8
1.23
1.45
3.0
2
4
4.06
2.93
1.5
-4
8
1.70
1.41
3.0
4
-2
1.12
1.09
1.8
4
0
1.11
1.45
3.0
-4
2
0.15
0.12
1.8
-6
0
2.21
2.56
3.3
-6
-2
2.29
2.06
1.8
4
4
4.84
5.37
3.6
-2
6
4.43
2.79
1.8
-6
6
3.13
2.73
3.6
0
6
0.46
0.66
1.8
-4
2
2.10
2.35
3.6
2
6
2.73
2.54
1.8
-4
-2
1.96
2.12
3.9
2
2
0.09
0.09
2.1
2
2
2.12
3.12
3.9
-4
-2
0.06
0.04
2.1
1 (m)
2 (m)
(sec.)
direction direction
Init
1 (m)
2 (m)
29
(sec.)