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
Summary
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
1
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 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 second-order
elastodynamic equation for a point in an elastic medium when perturbed by a
2
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
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
3
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.
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
4
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 space-time 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 semiinfinite 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 the construction of physicalmathematical formalism that supports the kinematic methods:
+∞
𝑢𝑖 (𝑥𝑗 , 𝑡) = ∫−∞ 𝑑𝜏 ∫Σ 𝐷𝑚 (𝜉, 𝜏)𝐶𝑚𝑛𝑜𝑝 𝜈𝑛
𝜕
𝜕𝜉𝑝
𝐺𝑚𝑖 (𝑥, 𝑡 − 𝜏, 𝜉)𝑑Σ
(1)
5
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  on the position  on the fault surface  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 finitesource 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
6
the time of rupture was discretized; l – sub-faults; m – components of the slip
vector (1 – strike direction, 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 3-D media. We used the code E3D, an explicit elastic finitedifference 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 [??]. The
usual regularization techniques [??] 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 non-negativeness 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 [??]. This
7
seems to be a unique reasonable method since as it is well-known [??] 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.
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 [??]. 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.
(1)
Here x is the unknown vector of amplitudes and residuals (see [??]) 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 ((1)) becomes a linear
programming problem. This observation is the key to an effective solution of
problem ((1)). 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 ((1)).
The following algorithm describes the process.
Algorithm:
Given 0 ,  > 0 , and  > 0 .
8
for k = 0,1,2,
Step 1. Solve linear programming problem ((1)) with  = k and obtain xk .
Step 2. Obtain search direction  k and a step  k > 0 .
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.
(2)
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 [??]) for problem
((2)) with  = 0 . By c B we denote the basic components of the vector c . If 
is sufficiently small, the solution to problem ((2)) 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.
(3)
Then we obtain
d
c, x( )
= cB , ( B( )) 1 B ( ) xB (0)
d
 =0
9


=  ( B( )) 1 cB , B ( ) xB (0) =  y( ), B ( ) xB (0).
T
This formula gives a possibility to calculate the gradient V ( ) of problem
((1)) value V ( ) . Indeed, taking  equal to i th basis vector, ei , we get
V ( )/i .
Thus, we obtain the search direction k = V (k ) and choose the step
k > 0
as
the
( xk ) B   ( B(k )) 1
maximal
 B (e )( x )
i
i
k B
 ]0, ]
satisfying
the
condition
 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,
(4)
where A is an (m  n) -matrix with m < n . Suppose that rank A = m .
Obviously x = 0 is an admissible point for problem ((4)). 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
10
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 )) -matrix. Consider a non-zero
vector
zk  R
xk = ( yk , zk )  R
n  mk
mk
R
and
n  mk
yk =  Bk1Ck zk  R
put
mk
.
The
vector
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
11
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, 1 and 2, where the rake angle of each source is
=atan(2/1). 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
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 finitedifference 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)
12
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 non-uniform 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 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 RootMean-Square Deviation (NRMSD) method to compare seismograms,
∑ (𝑢(𝑡) − 𝑢REF (𝑡))
𝑁𝑅𝑀𝑆𝐷 = √ 𝑡
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%).
13
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 start-time 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.
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; outros...)
revels an appropriate tool to make this since the scale of the problem does
not reach large dimensions (which size can we put Vladimir?). Is a inversion
14
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 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.
15
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).
16
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21
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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.
22
zj
1
u
-1
1
zi
Solution
space
-1
Fig. 2 – Geometric representation of DUAL formulation Solution
Fig 3 – Geometria da situação ensaiada
23
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
24
Fig. 5- Comparação entre o modelo de rupture original usado (esquerda) e o
reconstruído pelo método Simplex DUAL (direita)
Para a figura 5
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.
25
Fig. 6- à esquerda- Comparação de sismogramas produzidos pelo modelo real
(azul) e do modelo reconstruído (vermelho); à direita- distribuição total de
slip dos dois modelos (cima original) e baixo reconstruído obtidos com o
algoritmo Simplex DUAL
26
Fig. 7- Comparação entre o modelo de rupture original usado (esquerda) e o
reconstruído pelo método NNLS (direita)
27
Fig. 8- à esquerda- Comparação de sismogramas produzidos pelo modelo real
(azul) e do modelo reconstruído por NNLSv(vermelho); à direitadistribuição total de slip dos dois modelos (cima original) e baixo
reconstruído obtidos com o algoritmo NNLS
Fig. 9- Comparação da evolução temporal da taxa de momento (STF) dos três
modelos: o considerado (azul); o restaurado por programação linear
(vermelho) e o restaurado pelo método NNLS (verde)
28
Tabelas
Table 1
Position
in strike
axis
(km)
0
0
-2
2
0
4
-4
4
-2
0
-2
2
-2
2
4
-4
-6
-2
0
2
2
-4
Position
in dip
axis
(km)
0
-2
0
0
2
2
0
0
4
4
-2
-2
2
4
-2
2
-2
6
6
6
2
-2
Slip in
strike
direction
1 (m)
1.70
4.83
0.54
3.19
2.79
4.46
4.83
0.11
3.16
4.16
1.48
3.11
0.89
4.06
1.12
0.15
2.29
4.43
0.46
2.73
0.09
0.06
Slip in
-dip
direction
2 (m)
1.41
8.05
0.43
2.93
3.49
6.68
6.38
0.14
3.97
3.47
2.08
4.48
1.26
2.93
1.09
0.12
2.06
2.79
0.66
2.54
0.09
0.04
Init
time
(sec.)
0.0
0.6
0.6
0.6
0.6
0.6
1.2
1.2
1.2
1.2
1.5
1.5
1.5
1.5
1.8
1.8
1.8
1.8
1.8
1.8
2.1
2.1
Position
in strike
axis
(km)
-4
4
-6
-6
0
2
4
-4
-6
-4
-2
-6
2
-4
4
-6
4
-6
-4
-4
2
Position
in dip
axis
(km)
6
6
0
4
8
8
8
4
2
8
8
8
8
8
0
0
4
6
2
-2
2
Slip in
strike
direction
1 (m)
2.20
2.72
0.11
2.08
2.06
0.12
2.68
4.13
4.92
0.32
1.24
0.63
1.23
1.70
1.11
2.21
4.84
3.13
2.10
1.96
2.12
Slip in
-dip
direction
2 (m)
3.31
3.28
0.07
2.21
2.56
0.09
3.75
2.54
3.41
0.22
1.27
0.37
1.45
1.41
1.45
2.56
5.37
2.73
2.35
2.12
3.12
Init
time
(sec.)
2.1
2.1
2.4
2.4
2.4
2.4
2.4
2.7
2.7
2.7
2.7
3.0
3.0
3.0
3.0
3.3
3.6
3.6
3.6
3.9
3.9
29
MATERIAL SUPLEMENTAR
30
Para por na descrição do caso experimentado
We recall that simulations are only reliable at low frequency, the maximum
frequency being proportional to the minimum velocity in the grid, and to the
inverse of grid spacing. In this paper, a grid spacing of 1 km is used, thus
yielding a maximum frequency of 0.3 Hz. In the case of the two larger
earthquakes studied here (7.5 < Mw < 8.5), the finiteness of the dimension of
the fault and of the duration of rupture cannot be ignored. Following the
source implementation scheme of E3D, we model these extended sources by
superimposing a large number of point sources over a rectangular fault plane
that shares the same strike and dip as the individual subevents.
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Seim. Soc. Am. 72, 2037—2062.
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Hjorleifsdottir, Hiroo Kanamori, Thorne Lay, Shamita Das, Don Helmberger,
Gene Ichinose, Jascha Polet, and David Wald, 2005, Rupture Process of the
2004 Sumatra-Andaman Earthquake, Science 308, 1133-1139, DOI:
10.1126/science.1112260, 2005
31
[10] Bouchon, M. (1981). A Simple Method to Calculate Green’s Functions for
Elastic Layered Media, Bull. Seism. Soc. Am 71, 959—971.
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the Los Angeles fault system, Bull. Seismol. Soc. Am., 86, 575-596.
[12] Bao, H., Bielak, J., Ghattas, O., Kallivokas, L. F., O’Hallaron, D. R., Shewchuk,
J. R., and Xu, J., 1998, Large-scale simulation of elastic wave propagation in
heterogeneous media on parallel computers, Comput. Methods Apl. Meh.
Engrg., 152, 85-102
[13] Komatitsch, D., Ritsema, J., Tromp, J., 2002. The spectral-element method,
Beowulf computing, and global seismology. Science 298, 1737–1742.
[14] Hartzell, S. e Heaton, T. (1983). Inversion of Strong Ground Motion and
Teleseismic Waveform Data for the Fault Rupture History of the 1979
Imperial Valley, California, Earthquake, Bull. Seism. Soc. Am. 73, 1553—1583.
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in a Half- Space, Bull. Seis. Soc. Am. 82, 1018—1040.
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and Efficient Computation of Green’s Functions, Bull. Seis. Soc. Am. 89, 733—
741.
[17] Lawson, C. e Hanson, R. (1974). Solving Least Squares Problems,
Prentice-Hall (series in automatic computation), New Jersey.
[18] Ihmlé, P. F. (1998). On the Interpretation of Subevents in Teleseismic
Waveforms: The 1994 Bolivia Deep Earthquake Revisited, J. Geophy. Res. 103,
17919—17932.
[19] Hernandez, B., Shapiro, N. M., Singh, S., Pacheco, J., Cotton, F., Campillo,
M., Iglesias, A., Cruz, V., Gómez, J. e Alcántara, L. (2001). Rupture History of
September 30, 1999 Interplate Earthquake of Oaxaca, Mexico (Mw=7.5) from
Inversion of Strong-Motion Data, Geophys. Res. Lett. 28, 363—366
[20]
32
The synthetic Green’s functions necessary to simulate this data require the
use of 3D numerical wave propagation methods as finite differences or
spectral elements (referencias) that improved knowledge about the 3D
structure model in earthquake regions. – Isto é par air mais para a frente
quando se falar nas FG
The inverse problem consists to finding the values to source parameters
model that optimize the fits between the observed and synthetic data. The
choice of the method must be balanced against the ability of each method to
produce stable solutions for the specific problem to solve. If, hypothetically,
we want to estimate many sets of parameters (eg. mechanism and slip
distribution through space-time geometry), the situation is non-linear. If
there are ability to estimate sub sets of these parameters (eg. Caldeira et al.
2010) in order to restrict the number of parameters to estimate, could
transform it in a linear problem, solvable by linear inversion methods.
Since both the direct and the inverse problem require a numerical solution,
these techniques have been established as operational methods at around 70
years of the twentieth century, after the introduction of computers in field of
seismology, and the provision of data by the global seismic network.
The seismic source model usually considered, represents the rupture on a
rectangular surface (fault plane), oriented in a known geometry, usually
inferred from focal mechanism or other sources of geologic or geodetic
informations. The fault is divided into a set of N sub-faults (j = 1, 2 .. N)
distributed on a regular rectangular grid covering the entire surface. Each
sub-fault constitutes a point source which after a certain instant τj slips in a
defined direction, evoluting in time according the function D(t). The first slip
it happens on the sub-fault that contains the hypocenter, transmitting after to
neighboring sub-faults according a criterion of propagation that can be
defined simply by a uniform rupture velocity in all directions or according to
other criteria to be defined.
The model before presented assumes that each sub-fault releases all energy
at once, not returning to break. This configuration is known as the single
window model as opposed to another variant model of multiple windows presented by Olson and Aspel (1982) which allows that total slip of each subfault be distributed by different intervals.
The difference between the two model variants is in the way to represent the
propagation of rupture. The multiple windows version, thanks to the
possibility of break more than once, allows to simulate effects such local
variations of rupture velocity, stop phases and slip time variations to subfaults. However, this flexibility has a cost: the number of parameters to
estimate is increased by a multiplicative factor equal to the number of
33
windows inserted. Brian and Beroza (1994a) made a detailed description of
two models of rupture through a comparative study. In essence these models
are those that continue to be considered in more recent work of the
waveform inversion (eg Ichinose et al., 2003, Yagi et al. 2004; Vallée and
Bouchon, 2004).
The forword problem
The description of the elastic displacement produced at the Earth surface, as
consequence of applied forces or slip discontinuities in an elastic medium at
depth, constitutes the fundamentals for formal development of the methods
to seismic source study. The representation theorem in your integral form is
the starting point for the construction of physical-mathematical formalism relates that to
of the kinematic methods,
starting point for describing the seismic source, we use a representation where the forces are not
considered in volume and the tractions on the fault plane is supposedly where the continuous
modeling of the surface displacement is given by
ui (r,t) =
ò dt òò
t
S
is the popular form to
theorem for the earthquake source as a slip discontinuity in an elastic
medium
A representação do deslocamento, (eq. 2.1), é uma das equações mais
importantes da sismologia. A partir dela podem ser deduzidas as soluções de
vários problemas, com especial destaque para os relacionados com a fonte
sísmica. Na formulação mais popular mostra que os deslocamentos elásticos
produzidos em qualquer direcção i da posição x, ui(x, t), são explicáveis a
partir do efeito do deslizamento D (descontinuidade no espaço) de uma
parte da falha em relação à outra. Habitualmente, como ponto de partida da
construção do formalismo físico-matemático para descrever a fonte sísmica,
usa-se uma representação onde não são consideradas as forças de volume e
as tracções sobre o plano falha são supostamente contínuas (Aki e Richhards,
1980; Ben-Menahem, 1981; Udias, 1999; Pujol, 2003).
(Teorema da representação de Aki Richards na forma integral, onde inclua a
convolução)
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c, x  min ,
A( ) x = b,
x  0.
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