References

• Gallardo, L.A. and Maju, M.A. [2003] Characterization of heterogeneous near-surface

materials by joint 2D inversion of dc resistivity and seismic data. Geophysical Research

Letters, 30, 1658-1661.

• Gallardo, L.A. and Maju, M.A. [2007] Joint two-dimensional cross-gradient imaging of

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