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A note on the effects of material discontinuities on spatial variationsof surface displacements from static dislocations

Hasan Faik Kara a, Mihailo D. Trifunac b,n

a Department of Civil Engineering, Istanbul Technical University, Istanbul, Turkeyb Department of Civil Engineering, University of Southern California, Los Angeles, CA, USA

a r t i c l e i n f o

Article history:Received 21 December 2012Received in revised form23 September 2013Accepted 12 January 2014Available online 14 February 2014

Keywords:Asymmetry of static-surface displacementsGeological discontinuities in the vicinity offaults

a b s t r a c t

Post-earthquake observations show that permanent static and transient dynamic displacements in the nearfield of shallow faults can be complex and quite different from theoretical predictions for homogeneous half-space. These complexities are associated with many departures from the linear, elastic, homogeneous, andisotropic representations, and include the consequences of complex three-dimensional (3D) geologysurrounding faults. In this paper we illustrate these complexities for a two-dimensional (2D) model ofshallow strike-slip fault when it is inside, on the border, and outside a sedimentary basin.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Most earthquakes occur on the faults, with different rockmaterials in the two moving sides. This results from successiveslips during the active periods of faulting, and can be interpretedand quantified by geologic mapping [31]. Different effectiverigidities on the two sides of a fault then result in asymmetriesin both dynamic wave-radiation patterns and post-earthquakestatic deformations and creep. The purpose of this brief note is toillustrate the variations in the permanent deformations surround-ing a long strike-slip fault, which are caused only by differentmaterial rigidities. The emphasis will be placed on the variationsin the surface displacements, which is useful for the engineeringdesign of large and long structures that cross active faults (bridges,canals, tunnels, dams, large buildings, etc.).

The dislocation theory to determine static displacements causedby faulting has been discussed in the papers by Steketee [16,17],Rochester [14], and Kasahara [7] for a homogeneous elastic medium,and deformations on ground surface by three-dimensional (3D)plane dislocations were studied by Chinnery [3], for vertical faults,and by Mansinha and Smylie [12] for inclined faults. However, thefield observations show complexities and asymmetries in permanentground displacements around shallow and surface faults that cannotbe explained only by geometry and variable slip vectors on the faultsurface. For example, the northeast horizontal displacements in theTango 1927 earthquake in Japan were about 50 cm larger than the

displacement on the southwest side of the fault [30]. The displace-ments of the eastern side of the San Andreas fault, during the 1906San Francisco earthquake, were 0.5 m to 2.5 m smaller than thedisplacements on the west side of the fault [13].

The causes for the asymmetry of observed post-earthquakedisplacements have been studied by many authors. Ben-Zion [1],Rybicki and Kasahara [15], Mahrer and Nur [9], Mahrer [10], andHeaton and Heaton [5] wrote that it is important to considermaterial heterogeneity in the source region and argued their viewsin terms of geometrically simple models of linear media withdifferent rigidities. With the recent appearance of GPS and InSARmeasurements and with detailed modeling of three-dimensional(3D) inhomogeneities by numerical simulations, which also enableconsiderations of nonlinear material behavior, far more detailedand realistic inversions of the static displacement fields havebecome possible [2,6,11,4].

For the engineering design of long structures that cross the faults,it is necessary to select design values for possible surface-faultdisplacements and for the differential nature of motion across thefault line [18,24,25]. This has been done both deterministically andusing the hazard analysis [23,21,22], but assuming the symmetry ofmaterial properties across the fault surface. Since the forces instructures crossing the faults will depend on the differences in thenature and in the amplitudes of slip on the two sides of the fault, inthe following we illustrate this dependence for different ratios of therigidities on the two sides of the fault, and for the faulting inside,close to the edge, and outside the sedimentary basin.

The 2D model described in this paper is an approximaterepresentation of a dislocation along a long straight strikeslip fault. In future we will also search for a solution of the

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0267-7261/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.soildyn.2014.01.014

n Corresponding author. Tel.: þ1 213 740 0570; fax: þ1 213 744 1426.E-mail address: trifunac@usc.edu (M.D. Trifunac).

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corresponding plain strain problem, which is an approximation fora long thrust fault. The latter is mathematically more challengingbecause its corresponding dynamic solution (for P and SV waves)is not variable separable in the Cartesian coordinate system [8].

The complexities and 3D departures from simplified theoreticalpredictions of the surface displacements near faults can be alsoinferred from the associated observations of the dynamic radiationpatterns. Maps of peak and spectral amplitudes and of the associatedstrong-motion duration and energy all show complex but determi-nistic departures from the symmetric theoretical predictions ofradiation patterns in the elastic half-space [28,29,19,20,26,27].

2. Model

The model we consider consists of the semi-circular sedimen-tary valley, with radius a, surrounded by the elastic homogeneousand isotropic half-space (Fig. 1). The half-space is characterized bydensity ρs, rigidity μs and shear-wave velocity cs, while the semi-cylindrical valley is described by ρv, μv, and cv. The fault, whichexperiences permanent constant unit displacement across itswidth, is located at r¼ af , between the angles πþαf �αf l=2 andπþαf þαf l=2. The fault width is af αf l. When af ra, the fault isinside the valley and we examine the effects of the proximity ofthe fault to the edge of the valley. When af 4a, the fault is outsidethe valley, but the proximity of the valley to the fault introducesasymmetry in the permanent displacement field.

3. Solution

The governing equation for out-of-plane displacement incylindrical coordinates is

∂2

∂r2þ1r∂∂r

þ 1r2

∂2

∂θ2

� �uðr; θÞ ¼ 0: ð1Þ

Using the separation of variables solves the problemuðr; θÞ ¼ RðrÞΘðθÞ, and there follows

∂2

∂r2þ1r∂∂r

þ 1r2

∂2

∂θ2

� �RðrÞΘðθÞ ¼ 0; ð2aÞ

r2

RðrÞ∂2RðrÞ∂r2

þ rRðrÞ

∂RðrÞ∂r

¼ � 1ΘðθÞ

∂2ΘðθÞ∂θ2

: ð2bÞ

This equation holds if both sides are equal to a constant. If thisconstant is chosen to be n2, then

r2

RðrÞ∂2RðrÞ∂r2

þ rRðrÞ

∂RðrÞ∂r

¼ � 1ΘðθÞ

∂2ΘðθÞ∂θ2

¼ n2; ð3aÞ

∂2ΘðθÞ∂θ2

þΘðθÞn2 ¼ 0⇒ΘðθÞ ¼ C1einθþC1e� inθ : ð3bÞ

here, C1 is a complex constant and C1 is complex conjugate of C1.

r2∂2RðrÞ∂r2

þr∂RðrÞ∂r

�n2RðrÞ ¼ 0⇒RðrÞ ¼ B1rnþB2r�n; ð4aÞ

where B1 and B2 are real constants.Because uðr; θÞ ¼ RðrÞΘðθÞ,

uðr; θÞ ¼ ðB1rnþB2r�nÞðC1einθþC1e� inθÞ; ð5aÞ

uðr; θÞ ¼ ðC1B1rnþC1B2r�nÞeinθþðC1B1rnþC1B2r�nÞe� inθ : ð5bÞ

As the solution has to be prediodic in θ, n has to be an integer.The general solution that contains all integer values of n will thenbe a linear combination,

uðr; θÞ ¼ ∑1

n ¼ �1ðC1nB1nrnþC1nB2nr�nÞeinθþðC1nB1nrnþC1nB2nr�nÞe� inθ :

ð6Þ

This solution can be rewritten in a simplified form by combin-ing duplicated combinations as in,

uðr; θÞ ¼ ∑1

n ¼ �1ðC2nrnþC3nr�nÞeinθ : ð7Þ

It is convenient to use the absolute value of power r, as follows:

uðr; θÞ ¼ ∑1

n ¼ �1A1;nrjnjeinθþ ∑

1

n ¼ �1A2;nr�jnjeinθ ; ð8Þ

such that

A1;n ¼C2;n; n40C3;n; no0

( ); A2;n ¼

C3;n; n40C2;n; no0

( ); ð9Þ

Because the series converge, it is possible to approximate thesolution by a finite sum with N terms, as follows:

uðr; θÞ ¼ ∑N

n ¼ �NA1;nrjnjeinθþ ∑

N

n ¼ �NA2;nr�jnjeinθ : ð10Þ

When the fault is inside the valley (af oa), the displacementfield can expressed as follows:

uðr; θÞ ¼

∑N

n ¼ �NA11;nrjnjeinθþ ∑

N

n ¼ �NA12;nr�jnjeinθ ; roaf

∑N

n ¼ �NA21;nrjnjeinθþ ∑

N

n ¼ �NA22;nr�jnjeinθ ; af oroa

∑N

n ¼ �NA31;nrjnjeinθþ ∑

N

n ¼ �NA32;nr�jnjeinθ ; aor

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;:

ð11Þ

Because the displacement has to be finite at the origin andmust go to zero for increasing r, A12,n and A31,n need to be takenout from the solution, as follows:

uðr; θÞ ¼

∑N

n ¼ �NA11;nrjnjeinθ ; roaf

∑N

n ¼ �NA21;nrjnjeinθþ ∑

N

n ¼ �NA22;nr�jnjeinθ ; af oroa

∑N

n ¼ �NA32;nr�jnjeinθ ; aor

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;:

ð12Þ

For convenience, we introduce α1 ¼ πþαf �αf l=2; α2 ¼ πþαf þαf l=2 as new variables.

Fig. 1. Geometry of the problem.

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The boundary conditions are:Zero stress on flat surface:

μir

∂∂θuðr; θÞ

���� θ¼ 0θ¼ π

¼ 0; ð13aÞ

where μi is the shear modulus, ðμi ¼ c2i ρiÞ, and subscript i stands forv (in valley, roa) or s (in the half-space, r4a).

Continuity of stress and displacement on the interface betweenvalley and half-space gives

μv∂∂ruðr; θÞ

����r ¼ a�

¼ μs∂∂ruðr; θÞ

����r ¼ aþ

; ð13bÞ

uðr; θÞjr ¼ a� ¼ uðr; θÞjr ¼ aþ : ð13cÞ

Continuity of stress in the divided regions of valley is as follows:

μv∂∂ruðr; θÞ

����r ¼ a�

f

¼ μv∂∂ruðr; θÞ

����r ¼ aþ

f

: ð13dÞ

The displacement difference in divided regions of valley is

uðr; θÞjr ¼ a�f�uðr; θÞjr ¼ aþ

f¼ f ðθÞ; ð13eÞ

where f ðθÞ is a function that defines the displacement differencebetween angles α1and α2 (here assumed to be a constant equal toone), and continuity of displacement elsewhere.

To satisfy the zero-stress condition on a flat surface, anotherfault is introduced, symmetric w.r.t. the x axis. With this imaginaryfault, f ðθÞ function will be of the form,

f ðθÞ ¼ fH½θ�α1��H½θ�α2�gþfH½θ�ð2π�α2��H½θ�ð2π�α1�g ð14Þ

and its finite Fourier transform is

f n ¼12π

Z 2π

0f ðθÞe� inθdθ: ð15Þ

Fig. 2. Surface displacements for unit constant dislocation on the surface strike-slip fault (af π=16) with width af π=8, vs. dimensionless distance x=a, for different rigidities inthe valley and in the half-space, expressed by ratios of their respective shear-wave velocities cs=cv .

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f n ¼

i2πnðHð2π�α1Þðe�2iπn�e� iα1nÞþHð�α1Þðe� iα1n�1Þþþe�2iπnHðα1ÞðHðα1�2πÞðe2iπn�eiα1nÞþeiα1n�1Þþ�Hð2π�α2Þðe�2iπn�e� iα2nÞ�Hð�α2Þð�1þe� iα2nÞþþe�2iπnHðα2ÞððHðα2�2πÞ�1Þeiα2n�e2iπnHðα2�2πÞþ1ÞÞ; na012πðHð2π�α1Þð�α1þα1Hð�α1Þþ2πÞ�Hð�α1Þðα1þð2π�α1ÞHðα1�2πÞÞþ�Hð2π�α2Þð�α2þα2Hð�α2Þþ2πÞþHðα2Þðα2þð2π�α2ÞHðα2�2πÞÞÞ; n¼ 0

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;:

ð16Þ

where H is the Heaviside function, such that

HðθÞ ¼0; θo01; θZ0

( ): ð17Þ

Since πrα1oα2r2π, f n becomes

f n ¼i

2πnð�e� iα1nþeiα1nþe� iα2n�eiα2nÞ; na0α2 �α1

π ; n¼ 0

( ); ð18Þ

The amplitude of the fault dislocation is assumed to be one. Theinverse of the finite complex Fourier transform in Eq. (14) is then

f ðθÞ ¼ ∑N

n ¼ �Nf ne

inθ : ð19Þ

Substitution of the general solution into the boundary condi-tions gives the coefficients, as follows:

μv∂∂ruðr; θÞ

����r ¼ a�

¼ μs∂∂ruðr; θÞ

����r ¼ aþ

⇒ ∑N

n ¼ �NfA21;nðμvjnjajnj�1Þ

þA22;nð�μvjnja�jnj�1Þ�A32;nð�jnjμsa�jnj�1Þgeinθ ¼ 0; ð20aÞ

uðr; θÞjr ¼ a� ¼ uðr; θÞjr ¼ aþ ⇒ ∑N

n ¼ �NfA21;nðajnjÞ

þA22;nða�jnjÞ�A32;nða�jnjÞgeinθ ¼ 0; ð20bÞ

Fig. 3. Surface displacements for unit constant dislocation on the strike-slip fault, which does not break the surface (af π=12), with width af π=8, vs. dimensionless distancex=a, for different rigidities in the valley and in the half-space, expressed by ratios of their respective shear-wave velocities cs=cv .

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μv∂∂ruðr; θÞ

����r ¼ a�

f

¼ μv∂∂ruðr; θÞ

����r ¼ aþ

f

⇒ ∑N

n ¼ �NfA11;nðμvjnjaf jnj�1Þ

�A21;nðμvjnjaf jnj�1Þ�A22;nð�μv n af�jnj�1Þ

�� �einθ ¼ 0;

�� ð20cÞ

uðr; θÞjr ¼ a�f�uðr; θÞjr ¼ aþ

f¼ f ðθÞ⇒ ∑

N

n ¼ �NfA11;nðajnjf Þ�A21;nðajnjf Þ

�A22;nða�jnjf Þ� f ngeinθ ¼ 0: ð20dÞ

Since the above four equations have to hold for all θ, coeffi-cients of einθ have to be equal to zero independently for all integervalues of n between –N to þN, so that 8Nþ4 linear equations areobtained to calculate 8Nþ4 unknown constants.

A21;nðμvjnjajnj�1ÞþA22;nð�μvjnja�jnj�1Þ�A32;nð�jnjμsa�jnj�1Þ ¼ 0; ð21aÞ

A21;nðajnjÞþA22;nða�jnjÞ�A32;nða�jnjÞ ¼ 0; ð21bÞ

A11;nðμvjnjajnj�1f Þ�A21;nðμvjnjajnj�1

f Þ�A22;nð�μvjnja�jnj�1f Þ ¼ 0; ð21cÞ

A11;nðajnjf Þ�A21;nðajnjf Þ�A22;nða�jnjf Þ� f n ¼ 0: ð21dÞ

When the fault is in the half-space (outside the valley), thedisplacement field can be expressed in this form:

uðr; θÞ ¼

∑N

n ¼ �NA11;nrjnjeinθ ; roa

∑N

n ¼ �NA21;nrjnjeinθþ ∑

N

n ¼ �NA22;nr�jnjeinθ ; aoroaf

∑N

n ¼ �NA32;nr�jnjeinθ ; af or

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;: ð22Þ

Boundary conditions will be:

(1) Continuity of stress and displacement on the interfacebetween valley and half-space:

μv∂∂ruðr; θÞ

����r ¼ a�

¼ μs∂∂ruðr; θÞ

����r ¼ aþ

; ð23aÞ

Fig. 4. Surface displacements for unit constant dislocation on the strike-slip fault, which does not break the surface (af π=8), with width af π=8, vs. dimensionless distance x=a,for different rigidities in the valley and in the half-space, expressed by ratios of their respective shear-wave velocities cs=cv .

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uðr; θÞjr ¼ a� ¼ uðr; θÞjr ¼ aþ : ð23bÞ(2) Continuity of stress in the divided regions of half-space:

μs∂∂ruðr; θÞ

����r ¼ a�

f

¼ μs∂∂ruðr; θÞ

����r ¼ aþ

f

: ð23cÞ

(3) Displacement difference in the divided regions of half-space:

uðr; θÞjr ¼ a�f�uðr; θÞjr ¼ aþ

f¼ f ðθÞ: ð23dÞ

When a general solution is substituted into boundary condi-tions, the following equations are obtained:

A11;nðμvjnjajnj�1Þ�A21;nðμsjnjajnj�1Þ�A22;nð�jnjμsa�jnj�1Þ ¼ 0;

ð24aÞ

A11;nðajnjÞ�A21;nðajnjÞ�A22;nða�jnjÞ ¼ 0; ð24bÞ

A21;nðμsjnjajnj�1f ÞþA22;nð�μsjnja�jnj�1

f Þ�A32;nð�μsjnja�jnj�1f Þ ¼ 0;

ð24cÞ

A21;nðajnjf ÞþA22;nða�jnjf Þ�A32;nða�jnj

f Þ� f n ¼ 0: ð24dÞ

For the case when the fault is between borders of valley andhalf-space, the displacement field can be expressed in a simplifiedform, as follows:

uðr; θÞ ¼∑N

n ¼ �NA11;nrjnjeinθ ; roa

∑N

n ¼ �NA21;nr�jnjeinθ ; aor

8>>>><>>>>:

9>>>>=>>>>;: ð25Þ

In this case, boundary conditions are:Continuity of stress between valley and half-space:

μv∂∂ruðr; θÞ

����r ¼ a�

¼ μs∂∂ruðr; θÞ

����r ¼ aþ

: ð26aÞ

Displacement difference between valley and half-space:

uðr; θÞjr ¼ a� �uðr; θÞjr ¼ aþ ¼ f ðθÞ: ð26bÞ

When a general solution is substituted into boundary condi-tions, this set of equations are obtained:

A11;nðμvjnjajnj�1Þ�A21;nð�μsjnja�jnj�1Þ ¼ 0; ð27aÞ

A11;nðajnjÞ�A21;nða�jnjÞ� f n ¼ 0: ð27bÞ

4. Results

We show examples of surface displacements betweenx=a¼ �2 and x=a¼ 2 for the fault at af ¼ 0:5a; 0:75a and a (whenthe fault is inside the valley) and for x=a¼ �3 to x=a¼ 1 foraf ¼ 2a (when the fault is outside the valley). For x=a greater than2, the role of the proximity of the valley diminishes and thesurface-displacement field gradually becomes symmetric, withamplitudes that correspond to those in the elastic half-space.Consequently we do not show any figures for af 42a.

It can be seen from Figs. 2–4 that the presence of softermaterials inside the valley results in asymmetric surface displace-ments, which are larger on the side of the fault and include mostof the valley. This asymmetry then also leads to long-wavelength

rotations in the direction, which is away from the regions occupiedby softer materials. This rotation is evident from the largerpermanent displacements on the right side of the fault inFigs. 2–4, when the fault is in the left half of the sedimentarybasin.

In Fig. 2, we show the surface displacements for the faults withfault width (af π=8), which cut the free surface (αf ¼ π=16). In thisfigure, the surface displacements show a jump with a normalizedamplitude equal to one. The displacements are larger and decayslower with distance from the fault, on the side of the faultadjacent to softer geologic materials. On the side, which isadjacent to the edge of the valley and beyond, the displacementsare smaller and die out faster due to the increasing rigidity of therocks outside the sedimentary basin.

In Figs. 3 and 4 the fault is just below the surface, but does notcut the surface (αf ¼ π=12 and αf ¼ π=8 respectively). At x¼ af , thesurface displacement is not zero, but is positive because of theasymmetric material rigidities on the two sides of the fault.

As the fault depth increases (Figs. 3 and 4), the difference ofsurface displacements from one side to the other of the faultdecreases and the amplitudes become smaller. For αf ¼ π=2 thesedifferences would disappear and the surface displacements wouldbe small and symmetric relative to x=a¼ 0.

5. Conclusions

We used a simple 2D model of a strike-slip fault, cutting thesurface and buried but near the surface, to point out the asymme-tries in the surface displacements and the associated long- waverotations caused by the material inhomogeneities near the fault.The results show that for engineering estimates of permanentsurface displacements the rigidities of the geologic materialssurrounding the fault must be included in the calculations. Forthe realistic 3D geometries of the geology surrounding the faults,the results will be more complex but will maintain the asymmetryillustrated in this brief note.

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