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E041 ESTIMATING STATISTICAL PARAMETERS FROM TUNNEL

SEISMIC DATAA. KASLILAR 1,3 , Y.A. KRAVTSOV 2 , S.A. SHAPIRO 3 , S. BUSKE 3 , R. GIESE 4 and TH. DICKMANN 5

1Department of Applied Mathematics, Delft University of Technology,P.O. Box 5031, 2600 GA Delft , The Netherlands.

2Maritime University, Poland.3Freie Universitaet, Germany.

4

GFZ Potsdam, Germany.5Amberg Measuring Technique (AMT), Switzerland.

Summary

Estimation of the seismic velocity field by deterministic methods resolves large scale structures of the

medium. When the medium becomes inhomogeneous, the use of statistical methods can help to resolve

the small scale variations of the velocity field.

In this study, the travel time fluctuations of first-arrival P waves are studied in the framework of geometrical

optics for estimating the statistical parameters of the medium in the Gotthard Base Tunnel, Multifunctional

Station (MFS) Faido, Switzerland. In particular, the standard deviation of velocity fluctuations and the

inhomogeneity scale length in the horizontal direction are estimated.

Introduction

Information on statistical properties of inhomogeneities in the elastic medium can be necessary for estimat-

ing uncertainties of seismic images. This is especially important for inhomogeneities of a size at the limit of

the seismic resolution. Statistical properties of heterogeneities can be used in seismic inversion, combined

with geostatistical approaches. Moreover, statistics of heterogeneities might serve as a new seismic attribute

useful for building a bridge between seismic and lithological rocks classification.

Statistical inverse problems in reflection seismics were studied by Touati (1999), Iooss et al. (2000) and

Gaerets et al. (2001). Detailed analysis of traveltime statistics of reflected waves was performed by Kravtsov

et al. (2003a).

Kravtsov et al. (2003b) extracted statistical parameters from refracted waves and verified theoretical results

by numerical calculations. In the study here, a real data application of the method is carried out by using

tunnel seismic data. It is seen that some limitations arise in the application of the method. By applying the

method, the standard deviation of the medium fluctuations and the inhomogeneity scale length in horizontal

direction are estimated.

Method for estimating statistical parameters

The traveltime fluctuations of refracted waves are studied in the framework of geometrical optics. The

properties of the inhomogeneous elastic medium are characterized by the refractive index n(r) which isrelated to the wave velocity as n(r) = V0/V(r). V0 is the velocity near the earths surface and V(r) is thevelocity at position r. For the analysis of the travel time fluctuations, the following relation between the

travel time and the optical path (r) is used (Kravtsov et.al., 2003b):

t =(r)

v0=

1

v0

n[r(s)]ds, (1)

where r(s) is the ray trajectory and s is the arclength.A smoothly inhomogeneous random elastic medium is assumed and only first-order perturbations are con-

sidered. In a smoothly inhomogeneous random elastic medium the refractive index n can be represented asthe sum of an average (regular), n(r), and a random, n(r), part:

n(r) = n(r) + n(r). (2)

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Travel time, velocity and other parameters of interest can also be represented in the form of Eq.(2). In the

framework of the first-order perturbation theory the value v = v v is connected with n by the linearrelation

v = v2n

v0. (3)

Correspondingly, the travel time fluctuations are represented by

t =

v0=

1

v0

nds, (4)

which deals with integration of fluctuations n along an unperturbed ray.We consider a regular ray trajectory in a plane-layered medium with refractive index n and a constant veloc-ity gradient depending on the vertical coordinate z. We suppose that velocity fluctuations v are proportionalto average velocity v(z):

v(z) = v(z), (5)

where is a statistically homogeneous random field with a Gaussian correlation function and variance 2 .The variance of the traveltime fluctuations is represented by

2t (, ) = DJ(, ), (6)

where

D =

lz

2

H

v20

, (7)

and

J(, ) = B2/2/2

d

(B2 2)3/2

2(B2 2) + 2. (8)

The value ( = X/H) represents the dimensionless distance, where X is the end point of the ray andH is the ratio of the near-surface velocity, V0, to the velocity gradient, k; H = V0/k. The B parameter

is defined as B() =

1 + (/2)2. The ratio of the inhomogeneity scale length in the vertical directionlz to the inhomogeneity scale length in horizontal direction lx is represented by = lz/lx. The integralcalculated numerically for = 0.1, 1, 10 is presented in Fig.1. In the case of large offsets ( 1) thestatistical parameters of the medium 2 lz and = lz/lx can be extracted by using non-linear fitting betweenthe relation given in Eq.(6) and experimental data. By using the travel time covariance function which is

expressed in Kravtsov et al. (2003b) the inhomogeneity scale length lx can be estimated. Estimation oflx ensures to estimate lz and . For the application of the method considered here, a sufficient numberof medium realisations and large offsets are necessary. The numerical simulations for large offsets can be

found in Kravtsov et al. (2003b). If the offset () of the experimental data is not much larger than unityand there are not enough medium realisations, it is seen that some limitations arise in the application of the

method.In the case of small offsets ( 1), Eq. (8) can be approximated as J(, ) / and the non-linearrelation between 2t , , and given by Eq.(6) becomes

2t (, ) =

lz

2

H

v20

=

lx

2

H

v20

. (9)

Then it is not possible to estimate 2 lz and separately and only the quantity of 2

/ = 2

lx can be

extracted instead of2 lz and . For estimating lx, the zero cross intervals of the first derivative of the traveltime fluctuations are used. The lx value is estimated by averaging these intervals.

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Figure 1: J(, ) versus calculated numerically for = 0.1, 1, 10.

Application to the Tunnel Seismic Data

The theory is applied to the observed travel times of Multifunctional Station Faido of the Gotthard Base

Tunnel (Switzerland). The geology of the area consists of a Penninic Gneiss Zone of the Switzerland Alps.

The seismic measurements were carried out by GFZ Potsdam and Amberg Measuring Technique, Zurich.

The shots were recorded with three component geophone anchor rods which were placed in 2m deep bore-

holes. The shot and receiver group intervals were 1m and 10m, respectively. By using the results of the

tomography from Giese et al. (2002), the region where strong velocity variations are estimated were used

for the application of the method. For the selected region the considered numbers of shots and receivers are146 and 10, respectively. Although the real geometry has sparse sampling, the data were processed by using

reciprocity. Each shot-receiver group is considered as a medium realization.

The theoretical travel time curve for refracted waves are fitted to the observed traveltimes for each shot-

receiver group for estimating the velocity at the near-surface, V0, and the velocity gradient, k of the medium(Fig. 2a). The travel time fluctuations are calculated by taking the difference between the theoretical and

observed travel times (Fig 2b). The average near-surface velocity and the average velocity gradient were es-

timated as V0 = 5403m/s and k = 27.74s1 respectively, which gives H = V0/k 195m. The maximum

offset used in this study were 150m, the value was calculated as = X/H = 0.769.To estimate 2 lx, the variance of the travel time fluctuations are calculated by using the following equation

2t =< t2 >= 1N

Ni=1

ti(x)

2 . (10)

By using a linear fitting procedure between Eq. (9) and real data, 2 lx is estimated as 0.026m, (Fig.3).By using the first derivative of the average travel time fluctuations, the average of the zero cross intervals

are calculated to estimate the horizontal inhomogeneity scale length lx. The estimated result is found aslx = 13m. By using the estimated lx value in 2 lx, is estimated as 4.5%.

- 300- - 200- - 100 0 500

0.02

0.04

0.06

X(m)

t(s)

0 0.2 0.4 0.6 0.8- 0.0006

0

0.0008

g

t(s)

(a) (b)

Figure 2: (a) Observed traveltimes (solid line) fi tted to the theoretical traveltimes (dashed line) of refracted waves.(b) Traveltime fluctuations calculated by taking the difference between the observed and theoretical traveltimes.

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0 0.2 0.4 0.6 0.8

0

1 10- 7

3 10- 7

5 10- 7

7 10- 7

g

st

2 2

(s )

X

X

X

X

Figure 3: The variance of the average traveltime fluctuations (thin line) and the fi tted line (thick line) to this data.

Conclusions

In this study the statistical parameters; standard deviation of the fluctuations and the inhomogeneity scale

length in horizontal direction are estimated as = 4.5%, lx = 13m respectively. The average near-surfacevelocity of the medium is estimated as V0 = 5403m/s which is in agreement with the tomography resultsof Giese et al. (2002). Considering the dominant frequency for P waves (800Hz.), the corresponding wave-

length is 6.75m. The estimated inhomogeneity scale length is nearly twice of the wavelength, which allows

to connect the fluctuations to medium variations. As a further study, the estimated statistical parameters

will be used in the tomographic inversion of the seismic wavefield to improve the accuracy of the velocity

field estimation. Although there is no clear geologic evidence of the inhomogeneities in this order, the di-

mensions of objects like quartz and amphibolite lenses present in the area will be studied in more detail to

compare the estimated results with the real geology.

References

Gaerets, D., Galli, A., Ruffo, P. and Della Rossa, E., 2001. Instantaneous velocity field characterization

through stacking velocity variography, 71th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,

1-4.

Giese, R., Klose, C., Otto, P., Selke, C. and Borm, G., 2002. Investigation of fault zones in the Penninic

gneiss complex of the Swiss Central Alps using tomographic inversion of the seismic wavefield along tun-

nels, EGS XXVII General Assembly, Nice, France, 21-26 April.Iooss, B., Blanc-Benon, Ph. and Lhuillier, C., 2000. Statistical moments of travel times at second order in

isotropic and anisotropic random media, Waves in Random Media, 10, 381-394.

Kravtsov, Y.A., Muller, T.M., Shapiro, S.A. and Buske, S., 2003a. Statistical properties of reflection travel-

times in 3D randomly inhomogeneous and anisomeric media, Geophys. J. Int., 154, 841-851.

Kravtsov, Y.A., Kasllar, A., Shapiro, S.A., Buske, S.and Muller, T., 2003b. Extracting statistical parameters

of elastic medium from refraction traveltimes, submitted to Geophys. J. Int.

Touati, M., Iooss, B. and Galli, A., 1999. Quantitative control of migration: A geostatistical approach,

Mathematical Geology, 31, 277-295.