joint inversion of p and s traveltimes
DESCRIPTION
In order to maximize the amount of information taken from a seismic tomography survey, P and S-wave travel times can be used under the same joint inversion scheme. In this work two different approaches have been used.TRANSCRIPT
References
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magnetotelluric and seismic traveltime data for structural and lithological classification.
Geophysics Journal International, 169, 1261-1272.
• Moser, T.J., Nolet G. and Snieder R. [1992] Ray bending revisited. Bulletin of the
Seismological Society of America, 82, 259 – 288.
• Soupios, P.M., Papazachos, C.B., Juhlin, C. and Tsokas, G.N. [2001] Nonlinear three-
dimensional traveltime inversion of crosshole data with an application in the Area of
Middle Urals. Geophysics, 66, 627-636.
• Yi, M.J., Kim, J.H., Chung, S.H. [2003] Enhancing the resolving power of least-squares
inversion with active constraint balancing. Geophysics, 68, 931-941.
Joint Inversion of P and S traveltime tomography data
using Poison ratio and cross-gradient constraintsPetros N. Mpogiatzis1, Costas B. Papazachos1, Panagiotis I. Tsourlos1 and George N. Vargemezis1
1 Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece.
Abstract
In order to maximize the amount of information taken from a seismic
tomography survey, P and S-wave travel times can be used under the same joint
inversion scheme. In this work two different approaches have been used. The
first inverts the two data sets subject to constant velocities ratio constraint, while
the second uses the cross-gradients function constraint. In both inversion
schemes, additional regularization is used and spatially variable Lagrangian
multipliers are applied to the model parameters. The obtained solutions suggest
that joint inversion can lead to improved results, stabilize the inversion process
with both inversion strategies and reduce the non-uniqueness of the problem.
However, the constant velocities ratio method can create artifacts and reduce
the resolution when P and S-waves velocities are uncoupled and vary
independently. Therefore, it can only be applied when reliable a-priori
information about the investigation area ensures that the models parameters are
linked with a constant Vp/Vs ratio. On the contrary cross-gradient method is
more robust and does not require any a-priori information, yet its effectiveness is
increased when structural similarities do exist.
Figure 1. Target model 1 for (a) P velocities and (b) S velocities respectively
Figure 2. First and third rows show inversion results that corespond to target model 1
shown in figure 1, after 1st and 5th iteration respectively, for P-arrivals when applying
separate (a, g), constant ratio (b, h) and cross-gradients (c, i) methods. Second and fourth
rows show similarly inversion results after 1st and 5th iteration respectively, for S-arrivals
when applying separate (d, j), constant ratio (e, k) and cross-gradients (f, l) methods.
Figure 4. First and third rows show inversion results that corespond to target model 2
shown in figure 3, after 1st and 3rd iteration respectively, for P-arrivals when applying
separate (a, g), constant ratio (b, h) and cross-gradients (c, i) methods. Second and fourth
rows show similarly inversion results after 1st and 3rd iteration respectively, for S-arrivals
when applying separate (d, j), constant ratio (e, k) and cross-gradients (f, l) methods.
Figure 3. Target model 2 for (a) P velocities and (b) S velocities respectively
Introduction
The emergent need for more detailed and trusted information about the earth
has been the motivation for carrying out different kind of surveys on the same
target. Moreover, the reduction of the average cost of geophysical methods
has made such surveys affordable and meaningful. Although the results of
different kind of datasets are frequently compared and cross-correlated mainly
qualitatively, only recently effective and robust algorithms have been
developed that try to interpret the various kinds of data together in order to
extract more accurate and stable solutions. The term "joint inversion" describes
the ability to invert simultaneously a combined dataset to recover the
underlying models. In seismic tomography the combined dataset is the travel
times of the P and S-wave fronts and the models that produce these responses
reflect the P and S-wave velocities of the medium respectively. Usually both P
and S arrivals can be obtained with minimum additional effort; hence joint
inversion of these datasets offers an inexpensive and efficient way to improve
their interpretation. Therefore, a joint inversion algorithm of P and S travel
times, can extract more useful information from the existing - or easy to collect
- datasets and enhance the effectiveness of seismic refraction tomography
surveys.
1
Seismic forward problem
In the current work the seismic refraction forward problem is solved by the
combination of graph theory and Ray Bending method, in order to ensure high
accuracy but also satisfactory performance (Moser et al. 1992; Soupios et al.
2001). The two-dimensional model space is discreetized in nodes with slowness
values. Graph theory and the Dijkstra algorithm are used in order to extract an
initial rough raypath (shortest path) between the source and the receiver. This
approximate ray is further optimized with bending method in order to satisfy
Fermat's principle with the use of Brend – Fletcher – Reeves algorithm. Beta –
splines interpolation is used in order to create smooth rays but also to minimize
the needed points that represent the ray. Graph theory produces initial rays that
are close to the actual seismic rays (i.e. global minimum). That is essential
because it ensures that the optimization algorithm is going to converge to the
global minimum and not to some local one. When the final ray path for a given
source – receiver pair is extracted, then the travel time ti is computed as:
Where lj denotes the length of the jth segment of the ray along the path and Sj
is the average corresponding slowness value for that segment. Equation (1)
also provides the required derivatives matrix (Jacobian). Both P-wave and S-
wave forward problems can be solved using this scheme.
i j j
Path
t l S (1)
2
General inversion scheme
In this work, nonlinear Least squares method is used as the general inversion
method. Non linearity and limited data acquisition aperture usually turn
geophysical inverse problems to be ill-posed. To ensure stability and increase
the convergence rate, additional constraints and regularization, that generally
reflect physical properties or some kind of a-priori information for the model
parameters is applied to the inversion equations. This has as a result the
increase of residuals (data fitting error), so the amount of regularization must be
chosen carefully to balance the resolution loss. Seismic refraction linearized
inverse problem can be reduced to the following discrete (matrix) notation:
Where e is the residuals vector (discrepancy between observed and
calculated travel times), for the initial model parameters vector m0, J denotes the
partial derivatives (Jacobian) matrix, and Δm is the model perturbation vector. To
extract the optimum , the objective function to be minimized is:
Where λ is the Lagrangian multiplier and L denotes the constraint or
regularization about the solution to be obtained. If L equals I, equation (3)
becomes a ridge regression (Marquardt – Levenberg) problem. The minimum of
q with respect to Δm yields to the normal equations:
The use of a scalar Lagrangian multiplier implies uniform application to all
model parameters. In this paper we have used the Yi and Kim's Active
Constraint Balance (ACB) method. As they have shown (Yi and Kim, 2003) a
spatially varying Lagrangian operator can enhance the resolving power of least
squares method and simultaneously provide the desirable stability to the
inversion procedure. The corresponding normal equations become:
Where Λ is a diagonal matrix with the Lagrangian multipliers for each one of
the model parameters. The multipliers are distributed linearly in logarithmic scale
between two predefined extreme values and they are set for each parameter
automatically according the resolution matrix and spread function analysis.
Therefore, large values are assigned to parameters with low resolution and small
values to high resolved parameters.
e J m
( ) ( ) ( ) ( )T Tq e J m e J m L m L m
[ ]T T T J J L L m J e
[ ]T T T J J L ΛL m J e
(2)
(3)
(4)
(5)
3
Joint inversion
objectives and methodologies
The need to obtain the maximum amount of information about the earth and
the frequent presence of dissimilar data for the same investigation site has been
the impetus for combined interpretation of these different types of data. Although
a-posteriori cross correlation of the models that are extracted from the separately
analysis of the different data is common, the a-priori joint treatment of the data
can reduce the ambiguity or non-uniqueness of the interpretation. The main
subject of research for any joint inversion problem is the relation that correlates
the different physical properties of the underlying models and therefore the
different types of observation data together. P and S-wave velocities for a certain
medium are possible to be coupled or some times to vary independently. A joint
inversion scheme has meaning only if the underlying models have the same
general structure, and maximize its effectiveness when the different physical
properties are directly and unequivocally linked. Any aberrance to that usually
degrades the overall performance of these methods. In this work P-wave and S-
wave first arrival data are inverted together with two different inversion schemes
and the results are compared. In both methods we construct the extended
linearized problem by combining the two sets of data, unknowns and Jacobians
together.
The linking equations qjoint between the P and S underlying parameters are
also inserted as weighted extra lines into the extended system. This corresponds
to an objective function of the type:
Where, e is the extended residuals vector, J the extended Jacobian, and Δm
extended parameter perturbation to the initial approximation model.
In our first approach we regarded a direct relation between the different
parameters and more specific, that the velocities of P and S-waves are changing
such a way that their ratio tends to remain constant. This a-priori assumption is
frequently valid in a variety of cases and can be used as the linking constraint.
The formulation in a discrete domain is a system of linear equations of the form:
Where and stand for the velocities vectors of the P and S-waves relatively and σ
is their ratio. The above set of equations can be inserted into the initial
(extended) system by using a first order Taylor series expansion.
The second approach is structural and presumes that there is no explicit
mapping between the dissimilar model parameters. However the general
structure of the model is the same in P-velocity and S-velocity terms. That
means that the two different physical properties tend to change synchronized at
the same location (not necessary in the same way). An effective way to measure
the geometrical (structural) similarity of the models is the cross-gradient function
(Gallardo and Meju 2003; Gallardo and Meju 2004) given by:
The above function is minimized (approaches zero), either when the gradients
of the models are parallel, either when at least one of the gradients is zero. This
un-normalized constraint favors the models with same structures but at the same
time allows independent (not coupled) variance of the models reflecting geologic
boundaries that are distinct only regarding P or S-wave velocities separately. In
order to implement it into the inversion scheme it is expanded in first order Taylor
series around an initial model.
P P P
S S S
J 0 m e
0 J m e
int
( ) ( ) ( ) ( )
subject to: 0
T T
extended
jo
q e J m e J m L m Λ L m
q
int jo P S q m m 0
int 0jo P S q m m
(7)
(8)
(9)
(10)
4
Results and discussion
Tests with synthetic data showed that the application of joint inversion
schemes for P and S travel time data can significantly improve the inversion
results and reduce the non-uniqueness of the interpretation. In figures 1 and 2
the "target" models (homogeneous earth including a body of positive velocity
anomaly in figure 1 and the addiction of one negative anomaly body in figure 2)
are presented for both P and S velocities.
Receivers are placed in two boreholes (spacing 4m) and wave sources are
located on the surface every 2 meters. Especialy, in the case of figure 1, S-
arrivals are missing from the left borehole, so the resolving power of
independent S-arrival inversion for the high velocity target area is extremely
limited.
In figures 2 and 4, the results from separate (independent) and joint inversion
with the two examined approaches are shown for the target models of figure 1
and 3 respecively. VP/VS ratio regarded constant and close to the pre-assumed
value, while the initial models for the inversion process are in any case
homogeneous earth. In the case of the Target model 1 (figures 1 & 2), both joint
schemes produce better solutions than the separate (independent) inversions.
The constant VP/VS ratios method is more effective in this case, as expected,
while the cross-gradients method also gives adequate results. In the case of
target model 2 (figures 3 & 4) separate inversions and the cross-gradient
method produce the better and quite resemplant results. On the contrary,
Poisson ratio method fails especialy in P-velocities model witch is “infected” from
the S-velocities parameters.
In general, the cross gradients constraint function is more robust and case –
independent, while constant velocities ratio constraint becomes more effective
when the physical properties of the medium are actually connected in such way.
If P and S-wave velocities are uncoupled, the later introduces artifacts and
reduces the resolution of the inversion. The use of the cross gradient constraint
can avert this effect and should be favored when no a-priori information about
the investigation area is available or variation of the Vp/Vs ratio are expected. If
the P and S-velocities are linked but with different ratio than it is assumed cross-
gradients tend to remain effective, while constant Vp/Vs ratios strategy gives
either over or under estimations of the real parameters values.
5