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A short note on the inversion of tectonic stress fields
405
Tectonophysics, 100 (1983) 405-411 Elsetier Science Publishers B.V., Amsterdam
A SHORT NOm
- Printed in The Netherlands
ON THE INVERSION OF TECTONIC
STRESS FIELDS
WANG REN Department of Geologv, Peking University, Beijing (People’s Republic of China) (Accepted
November
4. 1983)
ABSTRACT
Wang, R., 1983. A short note on the inversion of tectonic stress fields. In: M. Friedman and M.N. Toksoz (Editors), Continental
Tectonics:
Structure, Kinematics and Dynamics.
Tectonophysics, 100: 405-411.
Works on global and regional stress fields are briefly reviewed. Questions tectonic
encountered
stress fields are listed as: the earth model, driving forces, distinguishing
stress. For the linear elastic case, a scheme making use of the superposition least squares is described
in computing
criteria and residual
principle and the method of
in terms of inverting the stress field of Eastern Asia under the action of the
Indian, Pacific and Phi&pine plates. For the non-linear case and the residual stress field the use of a trial and error scheme utilizing the historical tectonic movements
is briefly described.
INTRODUCTION
By tectonic stress is usually meant the stress deviated from a long-term equilibrium state due to gravity alone. Because of the rheological property of the earth media, the latter state is taken to be the lithostatic stress. It is also known as the standard state (Anderson 1951), i.e.: ,=u,,=a,=
J0
‘pgd.+pgz
0)
where z is the depth from the earth surface. To compute the tectonic stress field, the foremost question is about the earth model. Since the time scale for tectonic movements is of the order of lo6 years or longer, the model is quite different from that determined by the inversion of short-period movements. To establish an earth model for long-term tectonic movement is one of the present endeavors of earth scientists. It is generally believed that for periods less than lo8 years, the upper portion of the lithosphere, especially the upper 35 km, may be taken as elastic (Tureotte and Oxburgh, 1976; Murrell 1976). In such a case, the lateral inhomogeneity of the crustal structure and the ellipticity of the earth may no longer be neglected, one may no longer use a spherical symmetric model. oo40-1951/83/%03.00
Q 1983 Else&r
Science Publishers B.V.
The next problems
question
is about
investigated
field inversion,
by earth scientists
one assumes certain
tion to compute the computed
forces,
this is also one
(e.g. Forsyth
possible
driving
and Uyeda,
of the main
1975). For stress
forces as well as their combina-
a stress field and then picks out the more proper ones by comparing results with actual measurements.
The third suitability
the driving
question
is about
of the selected
earthquake fault-plane features (Scheidegger. The last question
the distinguishing
model
and
driving
criteria forces.
by which
This
to judge
is usually
solutions, on the analysis of tectonic structures, 1982) and all kinds of paleopiezometers. is about
the initial
or residual
the
based
on
geomorphic
stress state in the region. This is
usually not dealt with in the literature and will be treated here in the last section. For the global tectonic stress field, one of the earliest works is due to Vening Meinesz (1947). He considered a spherical elastic shell of 30 km thick with an incompressible fluid core and calculated the shear pattern due to the variation of rotation speed and to polar wandering. Richardson et al. (1979) regarded the entire lithosphere
as an elastic shell 100 km thick and subjected
at ridges and subduction
it to driving
zones as well as drag at the bottom
forces placed
of the plates. They used
a finite element method to calculate the stress field due to different ratios between these forces and used fault-plane solutions. in-situ stress measurements and strikes of stress-sensitive
geological
(1980) had later extended
There was more research the introduction
features
as the distinguishing
the work to estimate
work done on the regional
of the finite-element
method.
criteria.
Solomon
et al.
stress magnitude. Tapponnier
stress field, especially and Molnar
after
(1976) made
use of plane-strain slip-line field theory for the perfectly plastic body to investigate the displacement field resulting from the collision of the Indian and Eurasian plates. They used the trend distribution and the relative movement of the faults to be the main criteria in judging the correctness of their assumption. S.Y. Wang et al. (1980) investigated
the same problem
and also added these criteria and Indian
the seismicity
in the elastic case by using the finite-eIement and fault-plane
and arrived at a ratio between
method,
the tractions
plates. Villote et al. (1982) considered
tion of a rigid die into an incompressible
solutions
in North-China
to
from the Pacific, Phillipine
the same problem
viscoplastic
method region
medium
both in the plane strain and plane stress cases. Similar
as the indenta-
by the finite-element work was also done
in Japan (Otsuki, 1982) in which 24 models were used in the computation. INVERSION OF THE LINEAR ELASTIC CASE
In most of the works mentioned above, for each distribution of boundary traction a different model is used. However, for the elastic case, we may calculate only a few unit loading cases and then make use of the superposition principle and the method of least squares to get a best fit. This is done and illustrated by the preliminary calculation of a two-dimensional stress field in Eastern Asia below.
Fig. 1. Structural
framework
where fault offsets are known,
TABLE
stress field in Eastern
Asia; elastic case. H, C, Tare places
I, II, III are areas where stress states are known.
I
Unit loading Case
used in inverting
cases on the seven segments
Segment
F,
F,
shown in Fig. 1. Multiplying factor
Solution
to the present example
relative
resultant
size
force 2.74
N14”40’E
2.73
N44’26’E
1.00
N51’21’W
1.125
N56O18’W
1.34
N70”E
1.34
N70’E
1
Sl
1.00
0
et
(2, = 0.89
2
Sl
0
1.00
a*
a, = 3.40 2.45
3
S,
1.00
0
Q3
a3
4
s2
0
1.00
a4
a4 = 2.50
=
5
S,
1.00
0
a5
a,=l.OO
6
S.I
0
1.00
a6
a6 = 0.80 1.20
7
Ss
1.00
0
a7
a,
8
Ss
0
1.00
“s
as = 0.80
9
=
1.62
S6
1.00
0
a9
a9
10
s6
0
1.00
aI0
a,,
11
s7
1.00
0
%I
a,, = 1.62
12
s7
0
1.00
a12
a,2
= =
=
orientation
0.59
0.59
40x
The
structural
boundaries
framework
is shown
of the region are subjected
and zero traction
in the tangential
into seven segments,
in Fig.
I.
The
Western
to zero displacement direction.
Northern
in the normal
direction
boundary
is divided
The remaining
where $ and F, of the unit load may be specified
The twelve unit-loading
and
cases are listed in Table I. For example,
individually.
we shall apply the
first unit-loading
case, i.e. &, = 1.00, F,, = 0. acting on segment 1 with all the other segments free of traction. The displacement and stress distribution within the region are then computed, especially at the points j ( j = 1, 2, 3.. .m) where measured values can be obtained.
We shall denote
the offset value of the faults at point j due
in the region by (I,~,, u,,r,. T, ,.r,. According to to F,, by ui,; the stress components the superposition principle, if the load is increased by a factor al. these values will also increase by a factor to become a,~,,, (Y,u~~,,.“151,~ %Ta,l,. unit-loading cases will be applied in turn. With the multiplication
Now, each of the factor a, the offset
values at point j will be denoted by a, Us ,. the stress components at pointsj will be aru.Xr,, ayko,k,, akr,,k,. Let the measured values at pointsj be A, (be it offset value or the stress components), by the superposition principle, we have m equations of the type:
5 ~kuk,(orbxk,,uv~,,r\,,k,)=A,
(2)
(j=1,2...m)
X=1
If m = n, we may just solve the simultaneous
algebraic
equations
to get the I+‘s. In
the usual case, m > n. then let: n
c
(.j= 1,2...m)
a,~,,-A,=D]
(3)
k=l
and formulate: i
w,)*=fM
From
( af/aak)
(4) = 0 we obtain
the sense of least squares
n equations
to the measured
in term of (Ye. This yields the best fit in values. For the present
example,
we have
taken the offset values at points H, G, T, (Fig. 1) to be 13, 3.5, and 4.7 respectively (they are estimated from the slip rate in mm/yr, it is their relative values that are important here). They are three of the A,‘s. The stress states are given in regions I, 11, III, where the directions of maximum compressive stress are obtained from fault-plane solutions and the values of their principal stresses should have been estimated from observations. But, for example, they are quite arbitrarily assigned - 230). We have taken their relative here as ( - 100, -4OO),(-40, -lgO)and(-40, size according to the seismicity and magnitude of earthquakes in these regions as well as to their relation to the offset values given before. These stress components are transformed into rectangular coordinates and provide nine of the Ai’s. Thus altogether we have twelve A,?. By computing each of the twelve unit-loading cases, we
409
get twelve sets of eq. 2, The solutions of these simultaneous equations are given in Table I. The relative magnitudes of boundary traction, with that acting on segment 4 taken as unity, and their orientations are shown in the last two columns of Table I. The above example is given only for the purpose of illustration, a more detailed investigation is underway and will be presented in another paper. THE INVERSION OF THE NON-LINEAR CASE AND THE RESIDUAL STRESS FIELD
If the constitutive equation is not linear or we have to consider large deformation, then the superposition principle is no longer valid. The inversion becomes very difficult. However, since tectonic movement entails irreversible deformation, whether the media
is considered
are indeed
non-linear
further
as elasto-plastic and,
stress produced
or visco-plastic,
as a result,
by external
the constitutive
there will be a residual
traction
will be superposed
stress
equations state.
Any
on top of this. It has
a considerable effect on subsequent tectonic movement. Therefore, in the inversion of the stress field, it is quite important to find the residual stress field first. Inasmuch as the superposition
principle
no longer applies,
the trial and error method
is used.
Structural geology Seismotectonics I ___----I
Mechanical parameters
I
Boundary tractions
L --__,--,--,--_-,--_------_,------I 1 Compute tectonic stress field, adjust parameters such that degree of safety at the firet epicenter is very smell and other epicenters low c Streee releese at the fit epicenter in the sequence by lowering the coefficient of friction P end compute a new stress field
l
Y
i Compare the differences of the two streee fields with observations Strain energy releese . . . . Seismic wave energy Regions of seismic rick . . . . Areas of aftershock and minor quakes Fault offset . . . . Vestige, survey. Stress distribution . . . . in situ measurement, source mechanism I Pass Un~tisfa~ory, revision of parameters Stress release at the next epicenter in the sequence with ir at the 1 previous epicenter resumed original value but retain fault offset.
i Predict future seismic risk localities Fig. 2. Block diagram of a mathematical simulation of earthquake sequence in North China.
I
I 1
Fig. 3. Structural
framework
circles show the earthquakes
used in the simulation in succession
of earthquake
from 1303 AD-1976
sequence
in North
China.
Numbers
in
AD.
The residual stress field is to be recovered by making use of a succession of historical tectonic movements in this area. R. Wang et al. (1980, 1983), in an attempt to predict future seismically dangerous area after the Tangshan earthquake, proposed a way to recover the residual stress field due to earlier earthquake slips by utilizing
the wealth of Chinese
historical
earthquake
records.
had fault
The iterative
scheme used is shown in Fig. 2. They selected the 14 earthquakes of M > 7.0 in North China from 1303 A.D. to 1976 A.D. (Fig. 3) and simulated them by comparing the two equilibrium stress fields before and after the fault slips, as the fault is held together The change
by a static and a kinematic
in strain energy,
the seismically
coefficient
dangerous
of friction,
respectively.
area, the fault offset etc. are
compared one by one to the earthquake magnitude, aftershock area and surveyed data. The comparison is used as a guide to adjust the parameters involved. In simulating the next event of the sequence, the fault offsets of all previous earthquakes are retained which in turn yields the residual stress. However, in that simulation the residual stress found is not significant; it is believed that greater tectonic events have to be included to get a better result. Nevertheless, the scheme seems to be a workable one; it will also be applicable to recover the residual stress due to tectonic movements provided enough data concerning these events can be collected. Besides, to get a tolerable solution, better constraints on the mechanical properties of the earth materials and on driving forces are necessary.
411
ACKNOWLEDGEMENT
The example has been carried out by Mr. Gang Haihua. REVERENCES Anderson, E.M., 1952. The Dynamics of Faulting. Oliver and Boyd, Edinburgh, 206 pp. Forsyth, D.W. and Uyeda, S., 1975. On the relative importance of driving forces of plate motion. Geophys. J.R. As&on. Sac., 43: 163-200. Murrell, S.A.F., 1976. Rheology of the lithosphere; experimental indications. Tectonophysics, 36: S-24. Otsuki, K., 1982. Earth, 4(l): 7-14 (m Japanese). Richardson, R.M., Solomon, SC. and Sleep, N.H., 1979. Tectonic stress in the plates. Rev. Geophys. Space Phys., 17: 981-1019. Scheidegger, A.E., 1982. Principles of Geodynamics, Springer Berlin, 3rd ed,, 395 pp. Solomon, S.C., Richardson, R.M. and Bergman. EA., 1980. Tectonic stress: models and magnitudes. J. Geophys. Res., 8S(B12): 6086-6092. Tapponnier, P. and Molnar, p., 1976. Slip hne field theory and large scale cooti~~~ta~ tectonics, Nature, 264: 319-324. Turcotte, D.L. and Oxburgb, E.R., 1976. Stress accumulation in the lithosphere, Tectonophysics, 35: 183-199. Vening Meinesz, F.A., 1947. Shear patterns of the Earth’s crust. EOS, Trans. Am. Geophys. Union, 28(l). Vilotte, J.P. and Daignieres, M., 1982. Numerical modelling of intraplate deformation: simple mechanical models of continental collision, J. Geophys. Res. 87(B13): 10709-10728. Wang, R., He, G.Q., Yin, Y.Q. and Cai, Y.E., 1980. A mathematical simulation for the pattern of seismic transference in North China, Acta Seismoi. Sin., 2(l): 42-53 (in Chinese with English abstr.). Wang, R., Sun, X.Y. and Cal, Y.E., 1983. A math~mat~~l simulation of earthquake sequence in North China in the last 700 years. Scientia Sin., Ser. B, 26: 103-112. Wang, S.Y. and Chen, P.S., 1980. A numericai simulation of the present tectonic stress field of China and its vicinity. Acta Geophys. Sin., 23(l): 45-53 (in Chinese with English abstr.).
Tectonophysics, 100 (1983) 405-411 Elsetier Science Publishers B.V., Amsterdam
A SHORT NOm
- Printed in The Netherlands
ON THE INVERSION OF TECTONIC
STRESS FIELDS
WANG REN Department of Geologv, Peking University, Beijing (People’s Republic of China) (Accepted
November
4. 1983)
ABSTRACT
Wang, R., 1983. A short note on the inversion of tectonic stress fields. In: M. Friedman and M.N. Toksoz (Editors), Continental
Tectonics:
Structure, Kinematics and Dynamics.
Tectonophysics, 100: 405-411.
Works on global and regional stress fields are briefly reviewed. Questions tectonic
encountered
stress fields are listed as: the earth model, driving forces, distinguishing
stress. For the linear elastic case, a scheme making use of the superposition least squares is described
in computing
criteria and residual
principle and the method of
in terms of inverting the stress field of Eastern Asia under the action of the
Indian, Pacific and Phi&pine plates. For the non-linear case and the residual stress field the use of a trial and error scheme utilizing the historical tectonic movements
is briefly described.
INTRODUCTION
By tectonic stress is usually meant the stress deviated from a long-term equilibrium state due to gravity alone. Because of the rheological property of the earth media, the latter state is taken to be the lithostatic stress. It is also known as the standard state (Anderson 1951), i.e.: ,=u,,=a,=
J0
‘pgd.+pgz
0)
where z is the depth from the earth surface. To compute the tectonic stress field, the foremost question is about the earth model. Since the time scale for tectonic movements is of the order of lo6 years or longer, the model is quite different from that determined by the inversion of short-period movements. To establish an earth model for long-term tectonic movement is one of the present endeavors of earth scientists. It is generally believed that for periods less than lo8 years, the upper portion of the lithosphere, especially the upper 35 km, may be taken as elastic (Tureotte and Oxburgh, 1976; Murrell 1976). In such a case, the lateral inhomogeneity of the crustal structure and the ellipticity of the earth may no longer be neglected, one may no longer use a spherical symmetric model. oo40-1951/83/%03.00
Q 1983 Else&r
Science Publishers B.V.
The next problems
question
is about
investigated
field inversion,
by earth scientists
one assumes certain
tion to compute the computed
forces,
this is also one
(e.g. Forsyth
possible
driving
and Uyeda,
of the main
1975). For stress
forces as well as their combina-
a stress field and then picks out the more proper ones by comparing results with actual measurements.
The third suitability
the driving
question
is about
of the selected
earthquake fault-plane features (Scheidegger. The last question
the distinguishing
model
and
driving
criteria forces.
by which
This
to judge
is usually
solutions, on the analysis of tectonic structures, 1982) and all kinds of paleopiezometers. is about
the initial
or residual
the
based
on
geomorphic
stress state in the region. This is
usually not dealt with in the literature and will be treated here in the last section. For the global tectonic stress field, one of the earliest works is due to Vening Meinesz (1947). He considered a spherical elastic shell of 30 km thick with an incompressible fluid core and calculated the shear pattern due to the variation of rotation speed and to polar wandering. Richardson et al. (1979) regarded the entire lithosphere
as an elastic shell 100 km thick and subjected
at ridges and subduction
it to driving
zones as well as drag at the bottom
forces placed
of the plates. They used
a finite element method to calculate the stress field due to different ratios between these forces and used fault-plane solutions. in-situ stress measurements and strikes of stress-sensitive
geological
(1980) had later extended
There was more research the introduction
features
as the distinguishing
the work to estimate
work done on the regional
of the finite-element
method.
criteria.
Solomon
et al.
stress magnitude. Tapponnier
stress field, especially and Molnar
after
(1976) made
use of plane-strain slip-line field theory for the perfectly plastic body to investigate the displacement field resulting from the collision of the Indian and Eurasian plates. They used the trend distribution and the relative movement of the faults to be the main criteria in judging the correctness of their assumption. S.Y. Wang et al. (1980) investigated
the same problem
and also added these criteria and Indian
the seismicity
in the elastic case by using the finite-eIement and fault-plane
and arrived at a ratio between
method,
the tractions
plates. Villote et al. (1982) considered
tion of a rigid die into an incompressible
solutions
in North-China
to
from the Pacific, Phillipine
the same problem
viscoplastic
method region
medium
both in the plane strain and plane stress cases. Similar
as the indenta-
by the finite-element work was also done
in Japan (Otsuki, 1982) in which 24 models were used in the computation. INVERSION OF THE LINEAR ELASTIC CASE
In most of the works mentioned above, for each distribution of boundary traction a different model is used. However, for the elastic case, we may calculate only a few unit loading cases and then make use of the superposition principle and the method of least squares to get a best fit. This is done and illustrated by the preliminary calculation of a two-dimensional stress field in Eastern Asia below.
Fig. 1. Structural
framework
where fault offsets are known,
TABLE
stress field in Eastern
Asia; elastic case. H, C, Tare places
I, II, III are areas where stress states are known.
I
Unit loading Case
used in inverting
cases on the seven segments
Segment
F,
F,
shown in Fig. 1. Multiplying factor
Solution
to the present example
relative
resultant
size
force 2.74
N14”40’E
2.73
N44’26’E
1.00
N51’21’W
1.125
N56O18’W
1.34
N70”E
1.34
N70’E
1
Sl
1.00
0
et
(2, = 0.89
2
Sl
0
1.00
a*
a, = 3.40 2.45
3
S,
1.00
0
Q3
a3
4
s2
0
1.00
a4
a4 = 2.50
=
5
S,
1.00
0
a5
a,=l.OO
6
S.I
0
1.00
a6
a6 = 0.80 1.20
7
Ss
1.00
0
a7
a,
8
Ss
0
1.00
“s
as = 0.80
9
=
1.62
S6
1.00
0
a9
a9
10
s6
0
1.00
aI0
a,,
11
s7
1.00
0
%I
a,, = 1.62
12
s7
0
1.00
a12
a,2
= =
=
orientation
0.59
0.59
40x
The
structural
boundaries
framework
is shown
of the region are subjected
and zero traction
in the tangential
into seven segments,
in Fig.
I.
The
Western
to zero displacement direction.
Northern
in the normal
direction
boundary
is divided
The remaining
where $ and F, of the unit load may be specified
The twelve unit-loading
and
cases are listed in Table I. For example,
individually.
we shall apply the
first unit-loading
case, i.e. &, = 1.00, F,, = 0. acting on segment 1 with all the other segments free of traction. The displacement and stress distribution within the region are then computed, especially at the points j ( j = 1, 2, 3.. .m) where measured values can be obtained.
We shall denote
the offset value of the faults at point j due
in the region by (I,~,, u,,r,. T, ,.r,. According to to F,, by ui,; the stress components the superposition principle, if the load is increased by a factor al. these values will also increase by a factor to become a,~,,, (Y,u~~,,.“151,~ %Ta,l,. unit-loading cases will be applied in turn. With the multiplication
Now, each of the factor a, the offset
values at point j will be denoted by a, Us ,. the stress components at pointsj will be aru.Xr,, ayko,k,, akr,,k,. Let the measured values at pointsj be A, (be it offset value or the stress components), by the superposition principle, we have m equations of the type:
5 ~kuk,(orbxk,,uv~,,r\,,k,)=A,
(2)
(j=1,2...m)
X=1
If m = n, we may just solve the simultaneous
algebraic
equations
to get the I+‘s. In
the usual case, m > n. then let: n
c
(.j= 1,2...m)
a,~,,-A,=D]
(3)
k=l
and formulate: i
w,)*=fM
From
( af/aak)
(4) = 0 we obtain
the sense of least squares
n equations
to the measured
in term of (Ye. This yields the best fit in values. For the present
example,
we have
taken the offset values at points H, G, T, (Fig. 1) to be 13, 3.5, and 4.7 respectively (they are estimated from the slip rate in mm/yr, it is their relative values that are important here). They are three of the A,‘s. The stress states are given in regions I, 11, III, where the directions of maximum compressive stress are obtained from fault-plane solutions and the values of their principal stresses should have been estimated from observations. But, for example, they are quite arbitrarily assigned - 230). We have taken their relative here as ( - 100, -4OO),(-40, -lgO)and(-40, size according to the seismicity and magnitude of earthquakes in these regions as well as to their relation to the offset values given before. These stress components are transformed into rectangular coordinates and provide nine of the Ai’s. Thus altogether we have twelve A,?. By computing each of the twelve unit-loading cases, we
409
get twelve sets of eq. 2, The solutions of these simultaneous equations are given in Table I. The relative magnitudes of boundary traction, with that acting on segment 4 taken as unity, and their orientations are shown in the last two columns of Table I. The above example is given only for the purpose of illustration, a more detailed investigation is underway and will be presented in another paper. THE INVERSION OF THE NON-LINEAR CASE AND THE RESIDUAL STRESS FIELD
If the constitutive equation is not linear or we have to consider large deformation, then the superposition principle is no longer valid. The inversion becomes very difficult. However, since tectonic movement entails irreversible deformation, whether the media
is considered
are indeed
non-linear
further
as elasto-plastic and,
stress produced
or visco-plastic,
as a result,
by external
the constitutive
there will be a residual
traction
will be superposed
stress
equations state.
Any
on top of this. It has
a considerable effect on subsequent tectonic movement. Therefore, in the inversion of the stress field, it is quite important to find the residual stress field first. Inasmuch as the superposition
principle
no longer applies,
the trial and error method
is used.
Structural geology Seismotectonics I ___----I
Mechanical parameters
I
Boundary tractions
L --__,--,--,--_-,--_------_,------I 1 Compute tectonic stress field, adjust parameters such that degree of safety at the firet epicenter is very smell and other epicenters low c Streee releese at the fit epicenter in the sequence by lowering the coefficient of friction P end compute a new stress field
l
Y
i Compare the differences of the two streee fields with observations Strain energy releese . . . . Seismic wave energy Regions of seismic rick . . . . Areas of aftershock and minor quakes Fault offset . . . . Vestige, survey. Stress distribution . . . . in situ measurement, source mechanism I Pass Un~tisfa~ory, revision of parameters Stress release at the next epicenter in the sequence with ir at the 1 previous epicenter resumed original value but retain fault offset.
i Predict future seismic risk localities Fig. 2. Block diagram of a mathematical simulation of earthquake sequence in North China.
I
I 1
Fig. 3. Structural
framework
circles show the earthquakes
used in the simulation in succession
of earthquake
from 1303 AD-1976
sequence
in North
China.
Numbers
in
AD.
The residual stress field is to be recovered by making use of a succession of historical tectonic movements in this area. R. Wang et al. (1980, 1983), in an attempt to predict future seismically dangerous area after the Tangshan earthquake, proposed a way to recover the residual stress field due to earlier earthquake slips by utilizing
the wealth of Chinese
historical
earthquake
records.
had fault
The iterative
scheme used is shown in Fig. 2. They selected the 14 earthquakes of M > 7.0 in North China from 1303 A.D. to 1976 A.D. (Fig. 3) and simulated them by comparing the two equilibrium stress fields before and after the fault slips, as the fault is held together The change
by a static and a kinematic
in strain energy,
the seismically
coefficient
dangerous
of friction,
respectively.
area, the fault offset etc. are
compared one by one to the earthquake magnitude, aftershock area and surveyed data. The comparison is used as a guide to adjust the parameters involved. In simulating the next event of the sequence, the fault offsets of all previous earthquakes are retained which in turn yields the residual stress. However, in that simulation the residual stress found is not significant; it is believed that greater tectonic events have to be included to get a better result. Nevertheless, the scheme seems to be a workable one; it will also be applicable to recover the residual stress due to tectonic movements provided enough data concerning these events can be collected. Besides, to get a tolerable solution, better constraints on the mechanical properties of the earth materials and on driving forces are necessary.
411
ACKNOWLEDGEMENT
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