Geophys. J. R . astr. Soc. (1982) 69, 607-621
Inversion of field data in fault tectonics to obtain the
regional stress - I. Single phase fault populations:
a new method of computing the stress tensor
Jacques Angelier *
Laboratoire de Giodynamique, Dipartement de
Giotectonique, Universiti P.e t M. Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France
Albert Tarantola and Bernard Vdette Laboratoire d’Etude
Giophysique de Structures Profondes, Institut de Physique du Globe, Universiti P.
e t M. Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France
Stratis Manoussis CNRS, Centre de Calcul d e I’IN2P3, 1 I
Quai Saint Bernard,
75230 Paris Cedex 05, France
Received 1981 March 20
Summary. We attempt a general definition of the inverse problem of computing the components of the regional stress tensor from a set of field data
including the measurements of the strike and dip of several faults, and the
directions and senses of relative motion along these faults (as indicated by
slickenslides). In previous treatments of this problem, no experimental errors
could be taken into account except those in measuring the pitch of slickenslides; thus, errors in the orientation of the fault (strike and dip), which have
considerable practical importance, were neglected. In our work, all experimental errors in the field measurements are taken into account, so that the
agreement between the computed stress tensor and the set of field measurements can be rigorously checked.
1 Basic assumption
The problem of computing the regional stress tensor in fault tectonics by using field data
has recently received some attention in the literature. Using Bott’s (1959) theoretical
analysis, Carey & Brunier (1974) published the first practical attempt at determining the
best mean stress tensor for a given population of fault-slickenslide measurements. Various
improvements, developments and applications of the basic method were then made (Angelier
1975, 1979a, b; Carey 1976, 1979; Armijo & Cisternas 1978; Angelier & Goguel 1978;
Etchecopar, Vasseur & Daigniires 1981).
In addition to fault populations, other structures such as tension gashes, joints, stylolites
and Riedel shears provide indications on the regional stress. Because their interpretation is
well known (e.g. Price 1966), these structures will not be studied hereafter. On the other
*Present address: Tectonique Quantitative, Universitk P. et M. Curie, 4 Place Jussieu, 75230 Paris Cedex
05, France.
6 08
J. Angelier et al.
hand, a simple geometrical analysis may be sufficient to obtain the three principal directions
o f stress when particular fault systems are found (conjugate faults: Anderson 1942).
In this paper, we consider a general case, where any planar discontinuity in the rock may
act as a fault, the motion of which depends on a single common stress tensor. We assume
that all faults moved during a single tectonic event. Practically, polyphase fault populations
which include structures of different ages and mechanisms are commonly found in the field.
The mathematical study of such heterogeneous populations requires, in addition to the
method described hereafter, a cluster analysis applied to faulting; this will be dealt with in
greater detail in a subsequent paper.
The basic hypothesis in the present work is that, although the orientation of the fault
may be arbitrary (e.g. pre-existing faults or joints anisotropies), the striae (slickenslides)
on the fault plane, which are the trace of relative motion of the two sides, must be in the
direction and sense of maximum shear stress (i.e. in the direction of the projection of the
stress vector on the fault plane; see Fig. 1). This assumption was made explicitly by Bott
(1959). Obviously, the basic hypothesis implies that interactions between fault motions and
local variations of the stress pattern are neglected. Thus, in the first approximation, all the
faults which move during the same tectonic event are considered to be moving independently in accordance with a unique stress tensor.
More precisely, let T be the unknown regional stress tensor, and let n and s be the unit
normal t o the fault plane and the unit striae on the fault plane, respectively. The stress
vector a is, by definition of T:
a =T.n.
(1)
The components of u on n and s are respectively:
n .o = n . T . n
s . a =s . T .n.
Our basic assumption on direction and sense of striae may then be written as follows (see
Fig. 1):
T .n = (n .T n) n + (s .T n) s
i3a)
s . T . n z 0.
(3b)
+
T. n
Figure 1. Geometry of fault plane, slickenslides and stresses. FP, fault plane. n, unitary normal to FP. s,
unitary stria (slickenslide) on FP, indicating direction and sense of relative notion along the fault. T . n ,
stress vector acting on FP (see equation 1). (n * T * n)n, normal component of stress. (s T n) s, tangential
component of stress (shear).
Inversion offield data in fault tectonics
-
I
6 09
Condition (3a) expresses that n and s are consistent with T in direction, condition (3b) is
required by the consistency in term of sense of fault motion.
Associated assumptions (3a) and (3b) can be written in a simpler form. From equation
(3a) we successively obtain:
Then, using (4d) and (3b) we finally obtain:
s . T - n = + dll T . n 11'
-
(n . T -n)'.
(5)
We have shown that (3a-b) * (5). Let us show now that ( 5 ) is equivalent to (3a-b). It is
obvious that ( 5 ) * (3b). From ( 5 ) we can write:
(T .n)'
= (n
- T .n)' + (s .T .n)'.
(6)
As n and s are orthonormal (Fig. l), equation (6) corresponds to the Pythagorean theorem
and means that vector T . n is in the plane generated by n and s. In that case, the components
of T . n on the basis (n, s) can be obtained by the scalar product, and equation (3a) holds.
We have thus shown that the geophysical assumption of coincidence of the direction and
sense of striae with that of maximum shear stress is completely described by the mathematical condition ( 5 ) .
2 The parameters of the problem
To describe the orientation of the fault plane and striae, we use the three independent
angles strike ( d ) , dip ( p ) and slip ( i ) as shown in Fig. 2. This choice is quite logical since
d, p , and i are the angles that geologists directly measure in the field using compass and
Figure 2. Geometry of fault and conventional measurements. 0, x, y , z, reference axes. d , azimuth of dip
of fault plane. p , dip of fault plane. Thus, the normal to fault plane FP has a dip of (n/2- p ) in the
direction defined by (d + n). i, pitch of slickenslides, i.e. angle between horizontal direction of fault plane
(HH') and slickenslides, chosen as follows: 0 < i < n for a normal fault, n < i < 2 n for a reverse fault. This
implies that stria s indicates the motion of the lower side of the fault. Thus, 0 < i < n/2 or 3 n/2 < i < 2 n
for a sinistral fault and n/2 < i < 3n/2 for a dextral fault.
610
J. Angelier et al.
clinometer; furthermore, a measurement error in one of these three angles does not affect
the two other measurements (independent errors). The components of the unit normal
(chosen with n, 2 0) and striae (chosen on the conventional sense of motion as indicated by
Fig. 2) are then:
n, = sin d .sin p
ny = c o s d .sinp
n, = c o s p
(7)
s, = - sin i cosp sind t cos i cos d
sy = - sin i cosp cosd
-
cos i sin d
s, = sin i sin p .
The results of all our measurements thus consist in a list of all the values of strike, dip, and
slip, measured for all the faults, and the corresponding standard deviations of these data.
These standard deviations are inferred both from the precision of the instrument and from
field observations (quality of each measured fault plane and slickenslide). Table 1 shows an
example of such a list of measurements.
Let us now examine the parameters describing the stress tensor. Let T be a tensor verifying equation ( 5 ) for a given fault, with T’ another tensor defined by:
T ’ =t l . T + t 2 . I
where t l is any positive constant and t 2 .I is any isotropic tensor. It is easy to see that T’ also
verifies equation (5). This corresponds to the intuitive fact that neither adding an isotropic
tensor nor multiplying the tensor by a positive constant can alter the predicted sense and
direction of striae on any fault.
Table 1. Actual data used in Fig. 3. Values in degrees, as defined in
Section 3 of text.. ,k , number of each fault.
c:
-31.
7.
-45.
-48.
-43.
7.
7.
1%.
PO
56.
7.
7.
7.
70.
7.
7.
7.
7.
7.
hb.
73.
72.
-35.
7.
165.
-51.
7.
74.
77.
65.
52.
7.
H5.
7.
1.
71.
63.
5d.
-43.
I I*.
-41.
7.
7.
7.
7.
7.
7.
4
7.
7.
6
7
R
5
7.
7.
I.
7.
7.
7.
7.
7.
81.
97.
95.
91.
10R.
~.
109.
119.
120.
62.
109.
11.
1.
.
109.
111.
102.
12
7.
13
7.
7.
7.
7.
15
7.
7.
7.
7.
7.
67.
7.
40.
7.
7.
7.
112.
7.
64.
67.
7.
7.
7.
125.
7.
7.
7.
7.
-3h.
I‘+Y.
-43.
151.
-43.
7.
7.
32.
54.
67.
1.
I .
65.
Hh.
7‘4.
-5r.
7.
5Y.
1‘4Y.
7.
166.
-LO.
I.
7.
77.
..
IlV.
7.
67.
131.
r.
135.
“5.
I.
7.
7.
7.
56.
I.
7%.
I.
7.
I.
7.
Iril.
7.
”1.
7.
q5.
5
7.
7.
7.
7.
7.
5h.
51.
167.
96.
97.
R1.
57.
7.
I5Y.
I
2
3
63.
b5.
7.
1.
7.
7.
-61.
7.
7.
61.
-31.
112.
7.
7.
-413.
111.. . .
~
65.
70.
-44.
-43.
k
7.
7.
106.
7.
7.
-1R.
-37.
-30.
0;
‘ 0
7.
162.
lS3.
-20.
c,P
71.
10
I1
IZ
16
17
lb
19
20
21
??
c’?
14
r‘5
2t
27
?I!
29
30
31
3<
33
Inversion of field data in fault tectonics - I
61 1
As T‘ is a symmetric tensor, it has six degrees of freedom. It is then obvious that we must
limit ourselves to the computation of a tensor T with four degrees of freedom, because
tensors like T’ with six degrees of freedom can be obtained from equation (8) using arbitrary
values of tl and t 2 . It is always possible to choose t l and t2so that:
where T is a particular deviatoric stress tensor which will be called the reduced stress tensor.
It is easy to show that for numbers verifying equation (9), we can find a (unique) number ri,
(modulo 27r) such that:
TI1 = cos $
Tz2 = cos (ri,
+$)
We arrive then at the particular form of T suggested by Angelier & Goguel (1979):
cos $
T=(ff
Y
ff
Y
cos($+:)
p
0
cos ($
\
+?)
It is important to point out that all tensors T’ obtained from T using equation (8)have
the same directions of principal stress, and that the ratio:
where h l , X2 and h3 are the eigenvalues of T (with hl > h, L A 3 ) , is also the same for the
whole family T’ (The ratio (1 2) defines the shape of stress ellipsoid and verifies 0 d 9 < 1 ;
with @ = 0 when h2 = h3 and 9 = 1 when h I = h z ) .
The basic unknowns of our problem are thus the four parameters ($, a, 0,y) used in
equation (1 1). The directions of principal stress and the ratio @ will then be simply obtained
by diagonalization of T.
3 The inverse problem
Let ($o, (yo, Po, yo) be an apriori estimate (independent or our set of measurements) of the
deviator T, and let u$ , IT:, uc and uz be the standard deviations of this a priori estimate. If
we have no a priori information on T, we will accordingly take an infinite (very large)
a priori standard deviation.
Let (do,p o , io)k be data and (uf,4,
u;)~ standard deviations for the fault number k (k is
an index, not an exponent).
In the general case, no tensor T exactly verifies equation (5) on each fault. What we are
searching for is a tensor T (defined by ri,, a, 0,y) and a set of ‘corrected data’ (d, p , i)k
which are compatible, i.e. which exactly verify equation (5) for each fault. Clearly there are
an infinite number of solutions. For example, we could take arbitrary values for d k and p k ,
and take for ik values which are compatible with a given arbitrary tensor T.
22
J. Angelier et al.
612
Let us recall that compatibility is expressed by
.
s T n=
dll T .n 11'
- (n
+
T .n)'.
Among the infinite set of solutions verifying this equation, we will choose the solution
which minimizes the following sum:
where N is the number of faults. It is obvious that, if no a priori information on the stress
tensor is available (i.e. when u $ , u t , uf and uz tend to be infinite), then the criterion (13)
tends t o the classical non-linear least-square criterion.
Equation ( 5 ) and the minimization of S given by (13) state the inverse problem.
4 Solution of the inverse problem
Let us first use more compact and general notations. First we introduce the following
vectors:
Co is the covariance matrix of the a priori values no (under the hypothesis of independent
errors this matrix is diagonal and contains the respective variances of the elements of no).We
introduce a vector function f(n) in which the kth component is defined by:
fk(n) = sk - T .nk - 4 T . n k )(T . n k ) - (nk . T - n k ) '
(15)
where nk and sk are functions of (a,
p, i)k as stated in equation (7); k is an index.
With these notations, the solution of the inverse problem is simply defined by the minimization o f
s = (7T -no)' c,' . (n
'
-
no)
(16)
which belongs to the set of solutions of:
f(n) = 0.
(17)
Equations (1 6-1 7) are obviously identical with the two equations of Section 3.
This implicit non-linear least-squares problem has recently been solved (Tarantola &
Valette 1982); it can be demonstrated that the solution of (16-17) is obtained by the
iterative algorithm.
71,
+ 1 = no -t Co . F i . ( F , . Co . FA)-'
. [F, . (nn - no) - f(nn)]
where F is the matrix of partial derivatives
(see Appendix for the computation of these partial derivatives).
(18)
Inversion ofjield data in fault tectonics - I
613
When the solution n has been obtained with the required accuracy, the covariance matrix
of the solution, C, can be approxbately obtained by a formula involving a linear approximation (Tarantola & Valette 1982):
C = Co - Co. F * . ( F .Co .Ff)-'.
F . Co
(20)
where the partial derivatives are taken at point n.
5 Resolution and presentation of the results
The solution (18, 20) of the problem (16, 17) is a generalized solution in the sense that the
solution always exists independently of the number of equations (data). If the amount of
data is low, or if the data are of very poor quality (very great variances), the solution ($, a,
0,y) will be defined more by the a priori estimate ( $ o , ao,Po, yo) than by the data. On the
other hand with reasonably good data, ($, a,p , y) will be nearly independent of ($o, ao,
Po, yo) especially if the a priori standard deviations
ug, o{, ux) that have been chosen
are very large. In any case, as stated by Tarantola & Valette (1982), a good estimate of the
resolution of a parameter is obtained by the ratio:
(ut,
I=
a posteriori variance of a parameter
a priori variance of the parameter
This ratio, named indetermination estimator, verifies:
-
For a well-resolved parameter, I = 0; for a poorly resolved parameter, I 1. We will see from
our example in Section 6 that the four parameters ($, a , p,y) are well resolved by our data.
The four first components of the solution 71 obtained from equation (18) are the
a posteriori values for ($, a,p , y), which define the a posteriori reduced stress tensor.
Accordingly, the first four rows and four columns of the matrix C obtained from equation
(20) represent their variances and covariances. As ($, a, 0,y) have no intuitive meaning, we
express these results in term of the directions of the principal stresses, by diagonalization of
the stress tensor. Then, from the a posteriori covariance matrix of ($, a,p , y), we can compute the a posteriori errors in the principal stress directions. Finally we draw these directions
on the classical Schmidt projection, with the 95 per cent confidence regions (see Appendix
for more details).
6 Example with actual data
We have applied the method to various sets of fault measurements from Neogene and
Quaternary rocks of Southern Greece, Western Turkey and Baja California (Mexico). Two
examples are shown here, the first one in more detail since the data set has been already
used by one of us (Angelier 1979b) for illustrating two previous methods, the first by
'direct inversion' and the other by systematic four-dimensional exploration. Thus, the
reader may compare, if necessaly, the results obtained by using these methods and the
present one for the set of measurements of Fig. 3 (Angelier 1979b, p. 60).
Table 1 shows the measured values of (d, p , i ) on 33 faults which affect the Messinian
reefal limestones of Agia Varvara, central Crete, Greece. All angels i satisfy 0 s i s 180°,
hence all faults are normal (see Fig. 2). Fig. (3a) shows that the faulting pattern is very
simple as most faults are clearly conjugate. Furthermore, the method of right dihedra
(Angelier & Mechler 1977) defines compressional and tensional zones of compatibility
between all faults (0- and 100- values of Fig. 3c, respectively) in good agreement with the
J. Angelier et al.
614
goemetrical analysis in terms of conjugate shears proposed by Anderson (1942). Thus, it will
be easy to see whether the computed tensor is reasonable or not.
We assume an error of the order of 7" on all angular measurements. This includes an error
of the order of 1" for the instrument (a large-sized, rather precise one: 'Universal compass',
Topochaix, Paris), and the error due to both the irregularities of faults and the inhomogeneities of the deformation.
Although the error estimation is quite difficult, the clarity of the geometry of these faults
suggests that our assumption is rather pessimistic.
As we did not want to define a pn'on' constraints for the four parameters describing the
tensor, we have assumed large enough a priori standard deviations:
$0=
0
a$ = 4 n
= 100
ao= 0
0;
Po= 0
of) = 100
Yo= 0
0;
= 100.
The iterations were stopped when equations (17) were verified with sufficient accuracy,
i.e. when the sum of absolute values of the components off(n) became smaller than
This has been achieved after 30 iterations (other experiments with lo-'' do not change the
results significantly).
Note that the computation was first made using arbitrary values of ( J / o , a0, Po, yo).
These values define an a priori tensor which greatly differs from the a posteriori one (e.g.
zero values of equation 23). For most of the data sets that we have analysed, the final result
a0,Po, yo). This shows that, at least for
does not depend significantly on the choice of ($J~,
simple fault patterns, the choice of an aprion' tensor has but little effect on the final result;
although we have used as our starting point for the iterative process the apriori values (for
tensor parameters and for data) the number of iterations could be significantly reduced by
using the values indicated for the tensor parameters by the right-dihedra method.
The results of the inversion, with the a posteriori standard deviations, are shown in Table
2 . Let us recall that the tensor and the faults and striae of the table are compatible, in the
sense that they verify equation (5) exactly for all faults.
Focusing our attention on the components of the tensor, the values obtained are (to be
compared with 23):
$ = 1.7180
(Y
= 0.4805
0"
= 0.1208
P = -0.0546
op = 0.1512
-0.2597
oy = 0.0989.
=
\
d'=0.1084
I
The standard deviations have greatly decreased, hence the parameters have been well
resolved by the data.
More precisely, the a posterion' covariance matrix of these four parameters is:
0.01 18
-0.0098
-0.0125
0.0041
-0.0098
0.0146
0.0067
-0.0006
-0.0125
0.0067
0.0229
-0.0086
0.0041
-0.0006
-0.0086
0.0098
Inversion of field data in fault tectonics - I
20974
PHI
0.53
3100 3509
2500
615
0 DAUB0
0 P i
L 1 4
Figure 3. Analysis of a simple fault population (33 faults in messinian limestones, Agia Varvara, province
of Iraklion, Crete, Greece). Schmidt’s projection of lower hemisphere (M, magnetic north; N, geographic
north). For more detailed explanations, see Angelier (1979b). (a) Plot of measurements (fault planes and
slickenslides; all faults are normal). (b) Corresponding elementary axes, as used for focal mechanisms of
earthquakes: normal to fault plane (circle + cross) and slickenslide (simple cross), B-axis (asterisk), P- and
T-axes (black and white squares, respectively). (c) Result of rightdihedra method (Angelier & Mechler
1977). (d) Result of the computation of tensor described in this paper. The 5-,4- and 3-branch stars are
the axes of maximum, intermediate and minimum stress, respectively. Ellipses of confidence are also
plotted. Large black arrows show the direction of extension. Values of @ (equation 12 of the text) and
standard deviations of angles ( d - d o ) , ( p - p o ) , ( i - i o ) are also given. See also Tables 1 and 2. (e)
A posferiori values for measurements. This set of faults and striae are, by definition, perfectly compatible with the tensor of (d). All related values (d, p , i) are thus slightly different from those of (a) (see
Table 2).
6 16
J. Angelier et al.
Table 2. A posteriori values for data used in Fig. 3. Values rounded to the nearest degree. Definitions in Section 3. ud, u p and u' are the a posteriori standard deviations of d, p , i for each fault
number k . Note that faults 16,25 and 28 were eliminated (see text, Section 6 ) .
d
0-d
5.
6.
5.
6.
2.
71.
54.
4 .
68.
5.
-6.
4.
-I.
-33.
-43.
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-35.
136.
154.
152.
4.
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5.
8.
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-23.
-32.
4.
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-5.
5.
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lb@.
5.
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6.
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113.
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49.
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20
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P -Po
22
I3
24
25
2d
7.
2Y
30
31
32
33
Fig. 3 graphically summarizes the results (24) and ( 2 5 ) , after diagonalization of the stress
tensor defined by (9); from the covariance matrix (25) we have computed the 95 per cent
confidence regions for each axis.
It is easy to see on Fig. 3 that the orthogonal axes of maximum compressive stress,
intermediate stress and minimum stress are in good agreement with the corresponding axes
inferred from both the simple geometrical analysis in terms of conjugate shears (Fig. 3a, b)
and from the use of the right-dihedra method (Fig. 3c). Note that, however, the barycentres
of the compatibility areas of Fig. 3(c) have no rigorous mechanical significance, as pointed
out by Angelier & Mechler (1977). Furthermore, there is no significant difference between
the present result of Fig. 3(d) and that obtained by using another type of tensor computation (Angelier 1979a, b).
Nevertheless, the reason for proposing the present method is that it takes into account all
angular errors on (d, p , i), whereas previous methods considered only the error on i. Consequently, all authors implicitly assumed that the measure of the orientation of each fault
plane (i.e. d and p ) is perfect, which is not methodologically satisfying. In fact, error in field
measurements of the independent angles d, p and i are generally of the same order of
magnitude, so that the role of error on the orientation of fault plane (both d and p ) may
well be greater than that on the simple angle i . Furthermore, we wish to point out that the
computation time required by the present method is much smaller than for a systematic
four-dimensional exploration, even when restricted to compatibility areas (Angelier 1979b).
We emphasize, however, that one must be careful when applying the present method instead
of a systematic exploration, since anomalous results may be obtained, especially for very
irregular o r dissymetric fault patterns.
Fig. 3(e) shows the a posteriori values for fault parameters. By a simple comparison with
Fig. 3(a) (see also Table 2 as compared with Table l), we see that the rotations of fault
617
Inversion offield data in fault tectonics - I
planes and slickenslides required to obtain a perfect consistency between the data set and
the stress tensor, are small, and consistent with measurement errors.
While applying the method, one may decide whether ‘anomalous’ data are ignored or not
in the computation. Because the criterion that we used in equation (13) is a least-squares
criterion, thus giving more importance to anomalous values, it is better to eliminate the
faults with high values of
This implies that these structures either belong to another tectonic event which remained
undetected in the field, or are not consistent with the stress tensor for various reasons (e.g.
interactions between faults or presence of some continuous deformation). For the data set
171079 1099 2717
-11216 0 . M D l ,
PHI
0.76
D P I
3 12
’Lo ‘10 I D 15 1s 1s
5 5 0
s
ZOlJ
I
Figure 4. Analysis of a population of normal and strike-slip faults (Arroyo Montado, Santa Rosalia Basin,
Baja California Sur, Mexico). Legend as for Fig. 3 .
J. Angelier et al.
618
of Fig. 3 , three faults were thus ignored when we imposed the constraint Sk < 3. However,
in numerous cases, similar to that of Fig. 3 , the tensor finally obtained does not change
significantly when all faults are taken into account (i.e. no constraint on Sk).
AS is the case for all non-linear iterative methods, our method only converges if the nonlinearity is not too strong. If during the iterative process the normal to one fault approaches
a principal direction of the stress tensor the problem Seems to become excessively non-linear.
In addition, we have evident physical reasons to suspect that such a fault will not be related
to the tensor. The fault is ignored for the current iterative step. If a fault is ignored more
than n times (n = 5 ) it is totally eliminated from the data set, in order to ensure convergence.
This first example was extremely simple so that the result could be intuitively checked.
Fig. 4 illustrates a slightly more complex example of computation of stress axes for a
population of 22 faults measured in Pliocene sediments of the Santa Rosalia Basin, Baja
California, Mexico (Colletta & Angelier 1981). Normal faults and strike-slip faults are
mixed (Fig. 4a) but there is no qualitative reason to split this data set. Looking at the data
plot, it is difficult to localize the precise direction of extensional stress between WSW-ENE
and NW-SE. Both the right dihedra method and our computation of the tensor show that
this faulting pattern is consistent with a single stress system; the resulting direction of extension is WNM-ESE (Fig. 5d).
7 Conclusion
We have described a general method to obtain the regional stress tensor. The iterative
‘algorithm converges reasonably well. In a subsequent paper we will discuss the more general
problem of separation of tectonic phases. To apply this method to the computation of
regional stress using seismic focal mechanisms (instead of neotectonic data), supplementary
assumptions must be introduced, in order to distinguish the fault plane and the auxiliary
plane.
Acknowledgments
This work was Supported by the French CNRS (ATP GCodynamique, Theme ‘Tectonique
Intraplaque’, and RCP no. 264, ‘Probl5mes inverses’), and by the CNEXO (Contrat 79/5929).
M. Jobert, Le Pichon and J . C. de Bremaecker kindly read the manuscript. Contribution
IPG n o , 566.
x.
References
Anderson, E. M., 1942. The Dynamics of Faulting, 2nd edn, Oliver & Boyd, Edinburgh, 206 PP.
Angelier, J., 1975. Sur I’analyse de mesures recueillies dans des sites faill6s: 1’utilitB d’une confrontation
entre les mhthodes dynamiques et cinkmatiques, C. r. hebd. Skanc. Acad. Sci. Paris D , 281,
1805-1808 (erratum: ibid., 283, 1976,466).
Angelier, J . , 1979a. Determination of the mean principal stresses for a given fault population, Tectonophys., 56, T17-T26.
Angelier, J., 1979b. NCotectonique de l‘Arc EgBen, Spec. publ. SOC.gkol. Nord., 3 , 4 1 8 PP.
Angelier, J . & Goguel, J., 1979. Sur une mkthode simple de determination des axes principaux des
contrahtes pour une population de failles, C. r. hebd. Skanc. Acad. Sci. Paris D, 288,307-310.
Angelier, J . & Mechler, P., 1977. Sur une mkthode graphique de recherche des contraintes principales
Bgalement utilisable en tectonique et en stismologie: la methode des dibdres droits, Bull. SOC.
&ol. Fr., 7 XIX 6, 1309-1318.
Armijo, R. & Cisternas,<A., 1978. Un problkme inverse en microtectonique cassante, C. r. hebd. S k m .
Acad. Sci. Paris D , 287,595-598.
Bott, M. H. p., 1959. The mechanisms of oblique slip faulting, Geol. Mag., 96,109-1 17.
Inversion of field data in fault tectonics - I
6 19
Burg, J. P. & Etchecopar, A., 1980. Determination des sysdmes de contraintes liCs B la tectonique
cassante du coeur du Massif Central franqais: la region de Brioude (Haut-Allier), C. r. hebd. S b n c .
Acad. Sci. Paris D, 290, 391-400.
Carey, E., 1976. Analyse numerique d’un mod2le mecanique tlementaire applique 1’Etude d’une population de failles: calcul d’un tenseur moyen des contraintes B partir des stries de glissement, thkse
de 3e cycle, Universite de Paris-Sud, 138 pp.
Carey, E., 1979. Rechetche des directions principales de contraintes associees au jeu d’une population de
failles, Rev. G601. dyn. Ciogr. phyn., 21,57-66.
Carey, E. & Brunier, B., 1974. Analyse theorique et numerique d’un modkle mecanique Blementaire
applique B l‘ttude d’une population de failles, C. r. hebd. S b n c . Acad. Sci. Paris D, 279, 891 -894.
Colletta, B. & Angelier, J., 1981. Faulting evidence of the Santa Rosalia basin, Baja California Sur,
Mexico, in Geology of Northwestern Mexico and Southern Arizona, pp. 265-274, eds Ortliel, L. &
Roldan, J., University Nac. Auton. Mexico.
Etchecopar, A., Vasseur, C . & Daignikres, M., 1981. An inverse problem in microtectonics for the determination of stress tensors from fault striation analysis, J. struct. Geol., 3,s 1-65.
Price, N. J. 1966. Fault and Joint Development in Brittle and Semi-Brittle Rocks, Pergamon Press,
London, 116 pp.
Tarantola, A. & Valette, B., 1982. Generalized non-linear inverse problems solved using the least-square
criterion, Rev. geophys. Space Phys., in press.
Appendix
Using tensorial notations (sum over repeated Greek indices) equation (12) may be written:
fko)= s,“ ~ . p npk - J(T.~
n $ . (T.? n t ) - (dT . np”)’
~
(Al)
where T, s k and nk are functions of $I, a,p, y , d k , p k , and ik as defined in the text:
L T31
T32
T33
cos$
ff
Y
ff
cos($+:)
p
Y
P
COS($
+f,
1
nf = sin d . sin pk
sf = - s i n i k . c o s p k . sindk + cos i k .c o s d k
nt = cos d k . sin pk
sk - - sin i k .c o s p k . c o s d k - cos ik. s i n d k
n: = cosp k
s$ = sin ik . sin pk.
I
Using these notations, the derivatives of fk(n) with respect to the parameters defining the
stress tensor are:
J. Angelier et al.
620
where
and where R k denotes the square root in (Al).
For the derivatives of fk(r)with respect to the measured quantities one obtains:
afk - afk
ad'
an:
afk afk
-=-
apk
an!
n'
. sin
af
afk
as:
as:
- . sin ik . sin pk . sin d k t - . sin ik . sin p k . cos d k
f
+ afk . sin ik cospk
as';
afk - afk . (cos ik . cos p k . sin d k t sin ik . cos dk)
aik
as:
+ --af . (- c o i"~ . cos p k . cos d k t sin ik . sin d k )
as:
t
af
~
as$
. (cos ik . sinpk).
The derivatives of f' with respect to d', p', or i' are null for k
the right sides of (A6) are given by:
afk
~
an;
= T,,,
1
si -? (TPpTpyn t
- 2 T,, n:
+ 1. The partial derivatives in
n,k Tapnp")
The eigenvectors and eigenvalues of (A2) are easily computed: let us first put:
a = cos $
I
i 31
c=cos $+4-
h = a b c t20rPy-ap2-br2-ccw2
p
=&2
1
3
-t0 2 t 7 2
+ 3/4
I
Inversion offield data in fault tectonics
-
621
I
We obtain then the three eigenvectors A ' , A2 and A3 by:
and the coordinates of the corresponding orthonormed eigenvectors by:
Ui
X' =
J<u')2
t
(v')2
U'
y' =
t (W'y
J(U')Z
t (u')? t (w')2
Z' =-J(u')2
Wi
t (u')2 t ( W ' y
(A1 1)
ui and wi are solutions of:
ti',
(a - A) u i t aui t yw' = 0
olu' t
(b -h)u' t pw' = 0
yu' t pu' t (c -A)w' = 0.
In order to solve this system without numerical difficulties, we choose in each case two
independent equations among those three.
The covariance matrix Cv of the components (Xi,
Y', 2') of the normalized eigenvector
can be expressed in terms of the covariance CT matrix of (JI, (Y, 0,y) in the linear approximation (see text):
Cv
==
M -C, . M t
where
The partial derivatives in (A13) can easily be obtained from (A9) to (A12).