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TODAY ’ S LECTURE
- Fault geometry
- First motions
- Stereographic fault plane representation
- Moment tensor
- Radiation patterns
FAULT GEOMETRY
The fault geometry is described in terms of the orientation of the fault plane and the direction of slip along the plane. The geometry of this model is shown in Figure 1.
Figure 1. Fault geometry used in earthquake studies. [
∧
n isnormal vector of the fault plane. isslip vector which indicates the direction of
motion of hanging wall block. axis is in the fault strike so
∧
d
x 1 φ isstrike angle. Thedip
angle δ gives the orientation of the fault plane with respect to the surface. Theslip
angle λ gives the motion of the hanging wall block with respect to the foot wall block.
The motion is calledleft-lateral for λ = 0 ,right-lateral for λ = 180 ,normal faulting
for λ = 270 , andreverse or thrust faulting for λ = 90. Most earthquakes consist of
some combination of these motions and have slip angles between these values. Note that the basic fault types can be related to the orientations of the principal stress directions. Actual fault geometries can be much more complicated. Such complicated seismic events can be treated as a superposition of the simple events.
δ
λ
x 2 x^3
φ 1 x (^1)
n
Dip n
Slip angle angle
Strike angle
FAULT PLANE
. d= 0
d
North
Foot wall block
Figure by MIT OCW. [Adapted from Stein and Wysession, 2003]
04 May 2005
FIRST MOTIONS
The focal mechanism uses the fact that the pattern of radiated seismic waves depends on the fault geometry. The simplest method is the first motion, or polarity, of body waves. Figure 2 illustrates the first motion concept for a strike-slip earthquake on a vertical fault.
Figure 2. The relation between the first motion and fault geometry
The first motion is compression when the fault moves “toward” the station and
dilatation for “away from”. A vertical seismogram records are upward for compression
and downward for dilatation. A problem is that the first motion on actual fault plane is
the same as that on theauxiliary plane which is perpendicular to fault plane, so the first
motions alone cannot resolve which plane is the actual fault plane. This is a fundamental ambiguity in inverting seismic observations for fault models. We need additional geologic or geodetic information such as the trend of a known fault or observations of ground motion.
STEREOGRAPHIC FAULT PLANE REPRESENTATION
The fault geometry can be found from the distribution of data on a sphere around the focus. A stereographic projection transforms a hemisphere to a plane. The graphic construction is a stereonet (Figure 3).
[Adapted from Stein and Wysession, 2003]
Figure by MIT OCW.
04 May 2005
(a) (b) Figure 5. (a) The stereonet of different types of faults. (b) Focal mechanisms and some seismograms for three different earthquakes. Compressional quadrants are shown shaded.
In reality, we plot the points where rays intersect the focal sphere, so that the nodal planes can be found, considering the ray as compression (upward first motion) or dilatation (downward first motion). Figure 5-(b) illustrates the focal mechanisms and seismograms for three different earthquakes.
MOMENT TENSOR
To know the source properties from the observed seismic displacements, the solution of equation of motion can be separated as below
4
Left-lateral
on this plane
Right-lateral
on this plane
Thrust fault
Normal fault
Vertical dip-slip
Focal sphere side view
Focal sphere side view
Focal sphere side view
STRIKE-SLIP FAULT
DIP-SLIP FAULTS
Image removed due to
copyright consideration.
Figure by MIT OCW.
[Adapted from Stein and Wysession, 2003]
04 May 2005
u (^) i ( x , t ) Gij ( x , t ; x 0 , t 0 ) fj ( x 0 , t 0 ) r r r r = (^) (1)
u i is the displacement, is the force vector. The Green’s function gives the
displacement at point that results from the unit force function applied at point. Internal forces must act in opposing directions,
f j Gij
x x 0
f − f , at a distance so as to
conserve momentum (force couple). For angular momentum conservation, there also
exists a complementary couple that balances the forces (double couple). There are nine
different force couples as shown in Figure 6.
d
Figure 6. The nine different force couples for the components of the moment tensor.
We define themoment tensor M as
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
31 32 33
21 22 23
11 12 13
M M M
M M M
M M M M (^) (2)
M ij represents a pair of opposing forces pointing in the direction, separated in the
direction. Its magnitude is the product [unit: Nm] which is calledseismic moment.
i
j fd
M 11
M 21
M 31 M^32 M^33
M 22 M^23
M 12 M 13
Figure by MIT OCW. [Adapted from Shearer, 1999]
04 May 2005
0
0
M
M
M (8)
Principal axes become tension and pressure axis. The above matrix represents that x 1 ′
coordinate is the tension axis, T, and x 2 ′^ is the pressure axis, P. (Figure 7)
Figure 7. The double-coupled forces and their rotation along the principal axes. [
RADIATION PATTERNS
P-wave potential in spherical coordinate is given by
r
f
r
φ ( r , t ) = − f ( t^ − r /^ α) =− (^ τ) (9)
where α is the P wave velocity, r is the distance from the point source, and τ is time residual. Therefore, the displacement field is given by the gradient of the displacement potential u = ∂ φ/∂ r
τ
τ α
τ
( , )= ⎛^1 ( )^1 ( )
2
f
r
f
r
u rt (10)
The first term in the right hand side is near field displacement because of the decay as and the last term is far field displacement with the decay as. When we consider the relation between internal force and moment tensor given by equation (5), we can find that the near field term has no time dependence but the far field term has time dependence. The relations are given by
1 / r^21 / r
x
f M
~ ∂ , f^ ~ M ~ M &( t )
τ ∂ τ
Therefore, the near field term represents the permanent static displacement due to the source and the far field term represents the dynamic response or transient seismic waves that are radiated by the source that cause no permanent displacement. Figure 7
X 2
X 1
P axis
T axis X^2
X 1
X 2 X^1
Figure by MIT OCW. [Adapted from Shearer, 1999]
04 May 2005
represents the near and far field behaviors.
Figure 7. The relationships between near-field and far-field displacement and velocity.
In spherical coordinates, far field displacement is given by
( / )sin 2 cos
u r = 3 rM &^ t − r (12)
The first amplitude term decays as. The second term reflects the pulse radiated from the fault, , which propagates away with the P-wave speed
1 / r
M &( t ) α and arrives at
a distance r at time t − r / α. is called the seismic moment rate function or
source time function. Its integration form in terms of time is given by equation (7). The
final term describes the P-wave radiation pattern depending on the two angle
M &( t )
(θ , φ). At
, the displacement is zero on the two nodal planes. The maximum amplitudes are between the two nodal planes. Figure 8 represents the far-field radiation pattern for P-waves and S-waves for a double-couple source.
θ=φ= 90 o
t
M(t)
M(t) = u(t)
Near-field
Displacement
Far-field
Displacement
Far-field
Velocity
u(t).
Figure by MIT OCW. [Adapted from Shearer, 1999]
