The Current Distribution of Deformation in the Western Tien Shan from Block Models Constrained by Geodetic Data

Brendan J. Meade and Bradford H. Hager Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA

Abstract. We interpret Global Positioning System measurements of interseismic

deformation throughout the western Tien Shan in the context of a block model which

accounts for important geologic features (faults) and physical processes (elastic strain

accumulation.) Through this analysis we are able to quantify the amount of deformation

localized on active structures. In the central part of the belt the Djuarnarik fault zone

appears to be the most important thrust fault, accommodating nearly five millimeters per

year of north-south shortening across it. Conversely, the most widely recognized strike

slip fault in the region, the Talas Ferghana, is found to have very little of the previously

estimated right lateral motion.

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Introduction

In central Asia, just to the north of the Tarim basin and the Pamirs, and south of

the Kazak platform, lie a series of east west trending ranges, collectively known as the

Tien Shan. These features sprawl across the borders of China, Kyrgyzstan, Uzbekistan,

and Tadjikistan, reaching heights of over seven kilometers. The great elevation and relief

of these mountains, as well as an abundance of active seismicity, hint at their recent

creation and continued growth. The dominant east - west orientation of the ranges in this

region is consistent with their growth in the context of shortening associated with the

India - Asia collision (Molnar and Tapponier, 1975).

Detailed campaign style Global Positioning System (GPS) surveys of this region

have been carried out over the last eight years. The primary result of this effort has been

to produce a detailed velocity field of the western Tien Shan (figure 1). Previous analysis

of a preliminary velocity field quantified the amount of north - south shortening taking

place across the Tien Shan (Abdrakhmotov, et al., 1994). This work demonstrated the

importance of the Tien Shan in the context of the current kinematics of the India - Asia

plate collision, as up to half of the present day convergence rate (DeMets et al., 1990) is

accommodated in the western part of the range.

In this paper we develop a method for estimating not only the total amount of

shortening but also the distribution of shortening across the range. To do this we will

estimate interseismic deformation using a block model. This approach allows us to

include the effects of elastic strain accumulation due to the locking of faults. With the

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inclusion of this process, we are able to produce velocity fields that vary smoothly across

block boundaries. This allows us not only to develop a physically realistic model of

interseismic deformation, but also to estimate slip rates on faults from geodetic data.

Though our representation of the important faults is quite simple we are able to show that

the pattern of deformation seen in the GPS determined velocity field is compatible with

geologically reasonable fault locations. Further, the extent to which the observed GPS

velocities are compatible with our model leads us to speculate on tectonic implications.

Block Modeling Technique

By adopting a method that allows for the discretization of a region into a finite

number of blocks, we are able model interseismic velocity fields while incorporating

faults which are known to localize deformation. The elastic strain accumulation

associated with the locking of these faults is treated in some detail in our model, as this

effect is allowed to propagate, not only through a given block, but also across fault

boundaries. Thus, our model, while consisting of discrete translating blocks, produces

continuous surface velocity fields. Souter (1998) developed a detailed method for

dealing with the complex geometry of areas with both strike slip and thrust faulting. We

extend the work of Souter (1998) to allow for the inversion of geodetic data. To study a

region in this way we must do two things; establish the geometry of the block model for

the area we are concerned with, and then calculate the quantities we are interested in

learning about.

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We define the geometry of our model by dividing the area of interest into a

number of blocks, bounded by faults that are thought to be important on a regional scale

(Rubin, 1999, Hamburger, 1999). These blocks are treated as structures that endure little

permanent deformation through the time over which geodetic measurements have been

acquired. While the mountains themselves are manifestations of permanent deformation,

we consider their growth to be small over the period considered.

For the Tien Shan we have developed a simple system consisting of eight blocks

and twenty-three fault segments (figure 2). While the geometry is approximate, our

estimates concur with the mapped fault locations in the central part of the belt centered

on seventy-five degrees longitude. The four major thrust faults in this area are, from

north to south, Issykata, Djuanarik, Kadjerti River, and Aksai. To the west we have

faults bounding both the north and south edges of the Ferghana basin, as well as one

between the Kazak platform and the Choktal ranges. East of the central region we have

established faults which bound Issykul and then are allowed to run through the narrow

and high part of the belt. Faults which appear to not form part of a closed path (to the far

western and eastern edges) are treated as semi - infinitely long so that edge effects, due to

their finite length, are minimized. All of the faults in our model are locked to a depth of

fifteen kilometers. The dip of all thrust faults is estimated to be forty five degrees,

consistent with focal mechanisms from the Harvard CMT catalog and from the detailed

study of the aftershocks associated with the 1992 Suusamyr earthquake (Mellors et al.,

1997.) In the central part of the range the two northernmost faults dip to the south while

the southernmost pair are north dipping. While this may be a crude approximation of the

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real geometry of dipping faults, we believe that a simple possible geometry is consistent

with the best current estimates of how faults slope at depth. As well, large deviations

from this dip angle may be manifested by the inability of our model to satisfy the

observables. The only faults that are treated differently are the Talas - Ferghana and the

small northwest to southeast trending fault segment at the western edge of Issykata. Both

of these faults are treated as vertical.

Our technique ensures that the total amount of accumulated yearly displacement

from one point to another is path independent. That is, the path integral of any

component of velocity around any closed loop is zero.

v dli! = 0 (1)

We assume that the interseismic, or secular, velocities dominate the deformation

observed by geodetic techniques. Noting that there have been no large earthquakes

during the period over which geodetic measurements were made can validate this

assumption. Both the regional KNET (Vernon, 2000) and global Harvard CMT

earthquake catalogs demonstrate that though there has been ample seismic activity during

the past ten years, none of the earthquakes have been very large (Mw > 6) with the

exception of the Ms = 7.3 August 1992 Suusamyr earthquake (Mellors, et al., 1997). This

earthquake occurred before the GPS network returned results so we do not have to

consider the coseismic deformation associated with it. Though postseismic deformation

is not modeled, we see little evidence of its existence in the observed velocity field. This

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is not entirely surprising given that crustal thickness estimates of nearly fifty kilometers

(Roecker, et al., 1993, Cotton and Avouac, 1994) imply that depth of the seismic zone in

this region is not comparable to that of the lithosphere, thus precluding a substantial

postseismic asthenosphere response (Savage and Prescott, 1978.) However this does not

preclude viscous relaxation of the lower crust, though we see little evidence of this.

Over several earthquake cycles, the total motion at a point on a given block can be

represented as the sum of yearly interseismic and coseismic displacement.

! ! !v v vB I C= + (2)

where !vB

, !vI, and !v

C are the block (or geologic), interseismic, and coseismic yearly

displacements respectively (figure 3a). Specifically, the coseismic velocity is the

velocity due to the slip on all of the fault segments that define a block boundary. In our

case we are concerned with trying to model interseismic velocities so we rearrange (1) for

!vI:

! ! !v v vI B C= ! (3)

We can think of !vC

as the elastic deformation associated with a yearly coseismic slip

deficit. Both terms on the right hand side of (3) must now be written as functions of

block motions. The total geologic or block motion of a point can be written as

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! !v mB= ! (4)

where ! is an assignment matrix which indicates which block a given station is on, and

!m is a vector of the horizontal components of block motion. Relating fault slip rates to

surface deformation due to interseismic elastic strain accumulation, the second term on

the right hand side of (3) can be written as

! !v sC= ! (5)

where !s specifies the strike and dip slip rates on block bounding faults. Slip rates are

related to surface velocities through elastic dislocation theory. This allows us to calculate

the surface displacements (velocities) due to an arbitrary rectangular dislocation (fault

slip rate) in an elastic half space (Okada, 1985). In our case, we transform geographic

coordinates to Cartesian coordinates using a locally tangent Lambert conical projection.

The matrix ! contains the partial derivatives of these Green’s functions with respect to

slip rates. The elements of ! are given by

( )! ij S i jj

U s x= " #$! ! ! ! !

, , ,% & (6)

where ( )!…U are the Green’s functions for the displacement due to a rectangular

dislocation in an elastic half space, !s contains the slip rates, !xi gives the coordinates of

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the ith observation point, !! j specifies the fault geometry, j is a fault segment index, and ν

is Poisson’s ratio. The slip rates on a given fault are determined by the velocity

difference of the blocks that bound it (figure 3b). Thus slip rates are geometrically

related to block velocities by

! !s m= ! (7)

where ! is a rotation matrix which describes how relative block velocities decompose

into strike - slip and dip - slip components of slip on a bounding fault planes. Using

equations (4), (5), and (7) we can now write interseismic surface velocities in terms of

block motions as

( )! !v mI= !" #$ (8)

This formulation allows us to predict interseismic velocities at any point if block motions

are known. In general this is not the case, and block motions are unknown. However, we

can use geodetically determined secular velocity fields to estimate block motions, by

solving (8) for !m . More precisely, we estimate block motions by minimizing the

weighted L2 norm of the differences in station velocities between our model predictions

and the GPS observations. This enables us to write our model parameter estimates as

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( ) ( ) ( )! !m W W vest

T

GPS

T

GPSGPS

= ! !"#$

%&'

!!

( )* ( )* ( )*1

(9)

where the observed GPS velocities (Herring, 1999) have been substituted for interseismic

velocities. The weighting matrix, WGPS

, is the inverse of the partial data covariance

matrix. The partial data covariance matrix only includes the error estimates and

correlations for the north and east components of velocity at each site. It is possible to

generalize this matrix to include correlations between pairs of sites, however this has not

been done at this time. From (9) we obtain estimates of block motions, and are then able

to use our forward model to predict a best fitting model velocity field and the components

of slip on faults using equations 8 and 7 respectively. This method has been generalized

to include a priori slip rate estimates as well (see appendix A for details.)

Comparing the GPS and model velocity fields

Not every GPS site surveyed was included in our analysis. We only used sites

with velocity component error estimates of less than two millimeters per year. Further,

we eliminated sites that were believed to have had significant non-interseismic

contributions to their velocity estimates. In the end this left 147 sites. The GPS velocity

field for the Tien Shan is most commonly shown relative to station AZOK that is taken as

a proxy for the relatively stable Kazak platform. However several nearby sites show

systematic velocities, of small magnitude, towards the northeast (figure 1). In other

words, the Kazak platform does not appear stable relative to AZOK. In light of this we

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use the side ADAR as a reference station. This eliminates much of the systematic

velocity signal at nearby sites. Using these sites, and a weighting matrix based on their

one sigma error estimates, our inversion returns the block motion estimates given in table

1 and a χ2 / d.o.f. = 1.31. This statistic indicates that our error estimates should be scaled

to larger values or that our current model could stand to be made slightly more

complicated.

Our estimate of block translations proves capable of producing a velocity field

with many of the features observed in the GPS velocity field (compare figures 1 and 4).

The model velocity field appears qualitatively similar and somewhat smoother (as is

predicated by our technique) than the observed data, while still maintaining the dominant

north-south shortening signature. The most direct comparison of the ability of our model

to produce the GPS velocity field is shown in figure 5, where we show residual (observed

- model) velocities at each station. The residual velocities are quite small, with a mean

residual velocity magnitude of ~ 1.3 mm / yr. While there appear to be some systematic

residual velocities near the intersection of the northern edge of the Ferghana Basin with

the western extent of the Djuarnarik fault zone, this patterning is atypical of the residual

velocity field. The distribution of the components of the residual velocities show a

rather Guassian distribution and a near zero mean (figure 6).

The distribution of north - south shortening in the central part of the belt

The most significant predictions that we obtain from this technique are fault slip

rate estimates (figure 7). Again these estimates are calculated decomposing a relative

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block velocity vector into components of a fault slip coordinate system after equation 7.

In the central part of the range, north - south shortening is dominated by thrusting along

the Djuararik fault zone where slip rates range between 7 and 8.5 millimeters per year.

The dip slip rate on the Issykata fault zone bounding the Kyrgyz range to the north is

about half of that seen at Djuarnarik, consistent with partitioning estimates from geologic

field work (Thompson, 2000.) We have not included the Oinak-Djar fault zone in our

model due to the difficulty of separating its effect from that of the nearby Kadjerti River

fault zone. It is possible to better distinguish between the effects of different faults by

using the same model with shallower fault locking depths. Variations in slip rate along

strike can be introduced into our model through block rotations.

Perhaps the simplest method for looking at the current spatial distribution of north

- south shortening is to look at strain rate maps of !!NN

. We calculate the velocity

gradient tensor by differentiating the components of the velocity field after they have

been uniformly gridded with splines. Once the velocity gradient tensor has been

calculated it can be decomposed into strain (symmetric) and rotation (anti - symmetric)

rate tensors. We can compare the spatial distribution of north - south shortening based on

the observed GPS velocities (figure 8) to that based on model velocities (figure 9),

derived from our estimates of block motions. For the strain rate map based on GPS data

we see overall shortening, yet a large amount of spatial variability, especially near 75o

longitude. This is due to the combination of noise in the data and close station spacing.

Further, it is not obvious that strain is localized on any given structure west of Issykata.

However, the strain rate map based on the model velocity field shows just this. Again the

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Djuarnarik fault zone shows itself to be important, appearing as a linear feature with large

magnitude values of !!NN

. Our block model has allowed us to see this distinction as we

have effectively made this calculation based on a noise free model of the interseismic

velocity field.

In the end it seems clear that some caution should be exercised when looking at

strain rate maps. Unsmoothed geodetic data (figure 8) can yield results which may

require a tremendous amount of analysis to understand, while utilizing model velocity

estimates provides a method for directly interpreting strain rates in the context of a

geologic model.

The amount of strike slip motion on the Talas Ferghana fault

Perhaps the most surprising slip rate estimate that we obtain is that for the Talas

Ferghana fault. Previous analysis of offset river channels and dated organic remains

produced a late Holocene right lateral strike slip rate estimate of eight to sixteen

millimeters per year (Burtman et al., 1996, hereafter referred to as BSM). This is

substantially greater than the right lateral estimate of about one-millimeter per year that

we obtain for better than half of the length of the fault (see figure 7). Our estimate of slip

on the Talas Ferghana fault produces the surprising result that a small amount of left

lateral motion on the southernmost segment that we have considered. This is the result of

the similarities of the velocities for sites on either side of the fault at the most southern

latitudes within our network, as well as the rapid decrease in the northern component of

velocity for sites immediately to the south of the Kadjerti River fault. The widely

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disparate nature of these two slip rate estimates begs an explanation. There are a few

questions that we should ask in the search for an explanation of these apparently

incompatible measurements. What are the differences in how we estimate this quantity?

In what way might unmodeled permanent block deformation affect our results? Is it

possible that the quantity we are concerned with varies substantially in time?

The two different slip estimates were derived in vastly different temporal and

spatial scales. While BSM have measured actual displacements across the fault trace

(divided by some bounding age based on C14 dating of organic material in sag ponds to

yield a slip rate), we have estimated the amount of yearly fault slip that has not occurred.

Both techniques seem to be after the same quantity, even though the difference in

displacement measurements utilized in each case differs over three to four orders of

magnitude. It seems unlikely that our technique has proven inadequate in this case given

our previously demonstrated success in recovering geologically estimated slip rates from

geodetic data on vertical strike slip faults in southern California (Meade and Hager,

1999.) The extent to which our proposed block motions are valid over geologic time

scales is open to some amount of discussion as the presence of the ranges themselves are

evidence of permanent shortening. However allowing for additional deformation with a

block model would only tend to increase our slip estimates on faults, because they are the

only place that deformation can be localized in our model. Thus is seems that permanent

deformation of blocks cannot explain away the Talas Ferghana discrepancy. Perhaps the

different values are mostly a function of the length of time over which displacements

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were observed. While the carbon dating reported by BSM is on the scale of thousands of

years our GPS measurements span less than a decade.

The possibility that the Ferghana basin is rotating counter clockwise could

certainly increase our slip rate estimate for the southern Talas Ferghana fault. Some of

the residual velocities just to the west of the fault suggest that there may be an unmodeled

rotation (figure 5). Increased coverage in the Ferghana basin could certainly help to

determine whether or not this mode of deformation is important. We must consider the

possibility that the slip rate we are estimating based on GPS velocities is a slip rate

different from the geologically determined late Holocene effective slip rate we have been

discussing.

Conclusions

The overall success of our method for modeling interseismic GPS velocities is

shown most clearly in the residual velocity field (figure 5). These small residuals give us

reason to believe that we have captured the first order behavior seen in the observed

velocity field with the use of a geologically and physically reasonable model. From this

we are able to look at several measures of north - south shortening across the range. Both

slip rate estimates, based on our estimates of block motions, as well as strain rate maps

show that the Djuanarik fault zone accommodates the most shortening in the central part

of the range. Our slip rate estimates also show that the Talas Ferghana fault seems to

have only a small amount of right lateral slip on it. Strain rate calculations have been

shown to be useful tools for exploring geodetic data sets, in the context of a geologic

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model, in our case, allowing us to quantify the distribution of deformation across the

western Tien Shan from geodetic data.

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References Abdrakhmatov, K. Ye., et al., Relatively recent construction of the Tien Shan inferred from GPS measurements of present day crustal deformation rates, Nature, 384, 450 - 453, 1996. Bennett, R. A., Global Positioning System measurements of crustal deformation across the Pacific - North America Plate Boundary in southern California and northern Baja, Mexico, Ph.D. Thesis, MIT, 1995. Burtman, V. S., Skobelev S. F., Molnar, P., Late Cenozoic slip on the Talas Ferghana fault, the Tien Shan, Central Asia, Geol. Soc. Am. Bull., 108, 8, 1004 - 1021, 1996, Cotton, F., Avouac, J. P., Crustal and upper - mantle structure under the Tien Shan from surface - wave dispersion, Phys. Earth Planet. Inter., 84, 95 - 109, 1994. DeMets, C., Gordon, R. G., Argus, D. F., and Stein, S., Current plate motions, Geophys. J. Int., 101, 425 - 478, 1990. Hamburger, M., Personal communication, 1999. Herring, T. A., Personal communication, 1999. Matsu’ura, M., Jackson, D. D., Cheng, A., Dislocation model for aseismic crustal deformation at Hollister, California, J. Geophys. Res., 91, 12,661 - 12,674, 1986. Meade, B. J., and Hager, B. H., Simultaneous Inversions of Geodetic and Geologic Data for Block Motions in Plate Boundary Zones, Eos. Trans. AGU, 80, 46, Fall Meet. Suppl., 267 - 268, 1999. Mellors, R. J., Vernon, F. L., Pavlis, G. L., Abers, G. A., Hamburger, M. W., Ghose, S., Iliasov, B., The Ms = 7.3 1992 Suusamyr, Kyrgyzstan, Earthquake: 1. Constraints on Fault Geometry and Source Parameters Based on Aftershocks and Body - Wave Modeling, Bull. Seis. Soc. Am., 87, 1, 11 - 22, 1987. Molnar P. and Tapponnier, P., Cenozoic Tectonics of Asia: Effects of a Continental Collision, Science, 189, 4201, 419 - 426, 1975. Okada, Y., Surface deformation due shear and tensile faults in a half - space, Bull. Seis. Soc. Am., 75, 1135 - 1154, 1985. Roecker, S. W., Sabitova, T. M., Vinnik, L. P., Burmakov, Y. A., Golvanov, M. I., Mamatkanova, R., Munirova, L., Three - Dimensional Elastic Wave Velocity Structure of the Western and Central Tien Shan, J. Geophys. Res., 98, B9, 15,779 - 15, 795, 1993.

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Rubin, C., Personal communication, 1999. Savage, J. C., Prescott, W. H., Asthenosphere Readjustment and the Earthquake Cycle, J. Geophys. Res., 83, B7, 3,369 - 3,376, 1978. Souter, B. J., Comparisons of Geological Models to GPS Observations in Southern California, Ph.D. Thesis, MIT, 1998. Thompson, S. C., Personal communication, 2000. Vernon, F., Personal communication, 2000.

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Table 1. Block motion estimates

Block name East velocity

North velocity

East error North error

(mm / yr) (mm / yr) (mm / yr) (mm / yr) Kazak Platform 0.00 0.00 0.00 0.00 Kyrgyz Range 0.16 2.24 0.24 0.29

Issykul 1.55 5.28 0.42 0.45 Chaktal Range -1.20 3.17 0.41 0.46

Central Traverse Ranges 0.91 8.17 0.31 0.32 Naryn & At - Bashi Basins -1.40 9.21 0.49 0.62

Ferghana Basin -1.66 6.64 1.03 0.90 Southern and Kokshal Ranges 0.28 13.65 0.46 0.61

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Figure Captions Figure 1. GPS determined velocity field shown relative to the station ADAR on the Kazak platform. Error ellipses represent the 47% confidence intervals associated with the GPS observations. Figure 2. Thick lines show the block boundaries used in our model. Major faults, geologic, and geographic features are labeled. Figure 3a. Here we show a graphical representation of how we estimate interseismic velocities for the case of a vertical right lateral strike slip fault. The light gray line indicates the fault trace, and the arrows indicate the approximate direction and magnitude of surface displacement. We can solve this graphical equation for the yearly interseismic displacement. Figure 3b. Here we show how relative block motions are related to fault slip rates. The light gray line indicates the location of the faults, and the solid arrows indicate the approximate magnitude and direction of each block’s velocity. Thin arrows indicate the sense of motion on each fault. Figure 4. Model velocity field shown relative to the station ADAR on the Kazak platform. These velocities are calculated based on the block motion estimates from our inversion of the GPS determined velocity field. Note the similarity to figure 1. Figure 5. Residual (observed - model) velocity field shown relative to the station ADAR on the Kazak platform. Ellipses indicate the 47% confidence intervals associated the GPS velocity estimates. Modeled and observed velocities are most similar when the residual velocity vector at a station is small. Figure 6. Here we show the frequency distribution of the residual velocity components. The distribution is fairly Guassian and has mean near zero. The mean magnitude of the residual velocity components is ~0.9 mm/yr. Figure 7. Slip rate estimates for the faults included in our model are shown above. Dip-slip rates are shown as the upper value with positive values indicating closure. Strike-slip rates are shown as the lower value with positive values indicating left lateral motion. All values are mm/yr. Figure 8. Here we show contours of the !!

NN field based on the observed GPS velocity

field. This is measure of the localization of north-south shortening. Note that complicated structures are due to data noise and station spacing.

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Figure 9. Here we show contours of the !!NN

field based on the model velocity field. This is measure of the localization of north-south shortening. This velocity field is substantially smoother than the observed. Note that the Djuanarik fault zone appears as an east west trending structure.

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Figure 1.

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Figure 2.

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Figure 3.

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Figure 4.

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Figure 5.

26

Figure 6.

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Figure 7.

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Figure 8.

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Figure 9.