Computers & Geosciences 28 (2002) 775–781
DIPSLIP: a QuickBasic stress inversion program for analysing
sets of faults without slip lineations$
Tobore Orife*, Luis Arlegui, Richard J. Lisle
Laboratory for Strain Analysis, Department of Earth Sciences, Cardiff University, Cardiff, Wales CF10 3YE, UK
Received 10 May 2000; received in revised form 30 June 2001; accepted 5 July 2001
Abstract
A simple computer program is described for estimating palaeostress tensors from orientation data from a set of
fault planes. The computation is based on a novel technique that allows the tensor to be estimated in situations
where directions of slip on the faults cannot be determined, but where the senses of the dip-slip component of slip
on the faults are known. The new technique greatly broadens the scope of palaeostress analysis, permitting the
analysis of faults lacking slickenlines but exhibiting offsets of horizontal marker beds. r 2002 Elsevier Science Ltd. All
rights reserved.
Keywords: Structural geology; Palaeostresses; Stress quadric; Slip sense; Stress tensor
1. Introduction
Faulting is the brittle response of rocks to tectonic
stresses. The geometrical properties of faults and
their movements are thought to be controlled by the
nature of the active stresses. Half a century ago
Anderson (1951), using the Navier–Coulomb theory of
brittle fracturing, suggested how the orientation of faults
are controlled by the directions of the principal stress
axes. Geologists realised that if faults are governed by
stresses, then the orientation information collected from
faults in the field could be used to characterise the
palaeostress tensor. This process has become known as
stress inversion.
Stress inversion based on Anderson’s theory is limited
to a consideration of stresses responsible for forming the
$
Code available from server at http://www.iamg.org/
CGEditor/index.htm.
*Corresponding author.
E-mail address: orife@cardiff.ac.uk (T. Orife).
original fracture of the fault surface. It is known,
however, that fault slip often takes place along
favourably oriented pre-existing planes of weakness
rather than always on newly formed fracture surfaces.
Wallace (1951) and Bott (1959) developed a theory
relating the direction of slip on reactivated faults to the
imposed stress state. Their theory assumes that the faultslip vector is parallel to the direction of resolved shear
stress on the plane of weakness. They demonstrate that
on a given plane of weakness the direction of slip
depends on four variables; three of these describe the
orientations of the principal stress axes and the fourth is
f; the ratio of the principal stress differences
(f ¼ ½s2 s3 =½s1 s3 ). The Wallace–Bott hypothesis
forms the theoretical basis of most methods of stress
inversion in current use (see Angelier, 1994, for a
comprehensive review of methods).
The advantage of Bott–Wallace methods of stress
inversion is that they offer the potential of computing
four of the six components of the full stress tensor. On
the other hand, these methods require data on the
orientations of the slip directions as well as the attitude
of the fault surfaces. These requirements can be often
fulfilled in faults exposed at the surface, where lineations
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 9 9 - 1
776
T. Orife et al. / Computers & Geosciences 28 (2002) 775–781
on the fault plane can be used as indicators of the slip
direction. In numerous other cases, however, the slip
direction can be difficult or impossible to determine.
These include situations where
(1) faults cut poorly indurated sediments and have not
produced well-defined slip lineations,
(2) faults identified in geophysical logs from boreholes
where the slip direction cannot be deduced, and
(3) faults identified by seismic mapping where only the
fault plane orientation can be estimated.
The standard methods of stress inversion cannot be
undertaken with incomplete data of this kind. To
address this problem, Lisle et al. (2001) describe a
method of stress inversion that can be carried out in
such situations where slip lineations are lacking, but
where the sense of the fault’s dip-slip component is
known. This paper describes a computer implementation of the new method.
2. Determination of fault slip sense from separation
The intersection of a fault surface with a planar
marker produces a cut-off line. Slip along the fault
usually results in the separation of the cut-off lines in the
foot and hanging walls of the fault (Fig. 1a). Many
natural faults allow the measurement of separation;
either in a vertical section at right angles to the strike of
the fault (dip separation) or in a horizontal section
(strike separation). Unfortunately, the recording of
separation alone allows only very broad limits to be
placed on the direction of net slip vector; the latter need
not be perpendicular to the cut-off lines but can be as
3
F
much as 901 from that direction. In general, this could
mean that the sense of dip separation could contradict
the sense of dip slip (Fig. 1a). However, in the specific
case where horizontal beds are faulted (Fig. 1b), or at
least where the cut-off lines are horizontal, the sense of
dip separation always accords with the sense of the dipslip component. This fact means that, although slip
lineations may be lacking, the observed separation can
sometimes be used to provide vital information on the
dip-slip component of faults. This in turn provides data
for stress inversion.
3. Dip-slip sense and the stress tensor
Lisle et al. (2001) employed the concept of the
representation quadric for the stress tensor to derive
the theoretical relationship between the stress tensor and
the sense of dip slip on fault planes. They show that the
sense of dip slip relates to the way the normal stress (sn )
on a fault varies as the dip angle changes (Fig. 2). For
normal faults, the normal stress decreases as the dip gets
steeper whereas for reverse faults the normal stress
increases as the dip (d) increases, i.e. for normal faults
qsn =qdo0; whereas for reverse faults qsn =qd > 0:
4. Stress inversion using DIPSLIP.BAS
The calculation of the stress tensor from data
consisting of a number of faults with known dip-slip
senses is performed by program DIPSLIP.BAS.
This program carries out a search to find the stress
tensor that best explains the recorded senses of the
F
H
slip?
2
1
3
inclined
marker
plane
sense of
dip separation
(a)
slip?
1 2
sense of
dip separation
H
horizontal
marker
plane
(b)
Fig. 1. Faults cutting planar marker beds to produce cut-off lines in foot wall, F; and in hanging wall, H: Their separation is
compatible with many possible net slip vectors, of which three are shown. (a) Where cut-off lines are not horizontal, sense of dip
separation (normal) need not agree with dip-slip sense. For example, possible slip vector 3 has dip-slip component of reverse sense.
(b) Where cut-off lines are horizontal, sense of dip-slip component is always same as sense of dip separation.
T. Orife et al. / Computers & Geosciences 28 (2002) 775–781
H
REVERSE
NORMAL
NORMAL
dip increases
R increases
σn decreases
6 σn
is negative
6 dip
1
R= σ
n
REVERSE
Fig. 2. Determination of sense from normal stress variation
with dip of fault plane. Orientation of normal to fault plane is
represented by radius of representation quadric (ellipse),
whereas orientation of corresponding stress vector acting on
fault is indicated by line normal to the ellipse. Length of radius
is proportional to 1/O(normal stress). Normal faults are
characterised by decrease in normal stress with increase in
dip. Opposite is true for reverse faults.
measured faults. The computation involves the following stages:
(1) Input the data consisting of the dip azimuth, dip
angle and observed sense of each fault.
(2) Calculate the direction cosines of the fault normals,
and of the normal of the ‘‘shadow’’ of each fault
(faults with a slightly steeper angle of dip than the
measured one). We recommend from our experience that this shadow fault be about 0.051 steeper
than the measured one.
(3) Define a trial stress tensor by incrementing,
according to stereographic grid pattern, the following four variables:
(a) the plunge for s1 (the axis of maximum
compression)
(b) the plunge direction of s1
(c) the pitch of s2 within the s2 s3 plane, and
(d) the stress ratio, f:
(4) Calculate the direction cosines of the principal
stress axes, s1 ; s2 ; s3 :
(5) Calculate the direction cosines, with respect to axes
parallel to s1 ; s2 and s3 ; of the normals of the fault
planes and their shadows.
(6) Determine the normal stress values on the
fault planes and their shadows using Eq. 2–34 in
Ramsay (1967, p. 35) and, by comparing their
magnitudes, determine the expected sense of dip slip
(normal or reverse) on each fault (see previous
section).
(7) Determine the proportion of faults for which the
expected sense matches the observed sense. If this
777
proportion equals or exceeds some prescribed
value, store the trial stress tensor attributes. This
trial stress tensor represents a possible solution to
the stress inversion problem.
(8) Repeat the steps 3–7 until a full range of trial stress
tensors has been considered and all potential
solutions have been found.
5. Data and results files
The program reads the data on fault orientations and
senses from a text file. This file should be given the
extension .TXT. Each line of the file specifies three
parameters for a single fault: the dip direction, the angle
of dip, the sense of dip slip (1 or 1 depending on
whether normal or reverse sense). These three items of
data are each separated by spaces or commas.
DIPSLIP.BAS creates four results files. They contain
different attributes of all the palaeostress tensors that
successfully predict the senses on the measured faults.
The files having extension .S1, .S2, and .S3 contain the
trends and plunges of s1 ; s2 ; and s3 axes, respectively.
The file with extension .PHI stores the corresponding
stress ratios (f).
A VisualBASIC version of the computer program is
obtainable on request from the authors.
6. An assessment of the solution percentage using
DIPSLIP.BAS: is it a fair and stable description
of the inversion results?
Lisle et al. (2001) propose the solution percentage as a
possible measure for determining the precision (or
spread) of the results produced. They define the solution
percentage as the proportion of obtained solutions to
the total number of tensors tested. This measure was
employed in their publication to investigate a variety
of issues affecting the analyses including: the effects
of a preferred orientation of the fault data, varying
sample sizes of fault datasets and poor data quality.
However, they do not discuss the presumed stability or
fairness of such a measure. We have conducted an
experiment using DIPSLIP.BAS to test the presumed
fairness and stability of the proposed solution percentage measure.
The grid search parameters determine the number of
trial stress tensors that are tested in the inversion.
Firstly, tests were undertaken to ensure that irrespective
of the mesh size for the grid search selected, there is no
apparent bias in the stress axes orientations generated by
the algorithm that is implemented in DIPSLIP.BAS.
The fairness of the solution percentage measure was
then assessed by observing the number of solutions
produced by an inversion of two test datasets (listed in
778
T. Orife et al. / Computers & Geosciences 28 (2002) 775–781
Table 1
Test fault-slip dataa
A
Dataset A
051
049
010
Dataset B
354
307
157
185
146
117
182
206
221
347
B
C
10
19
44
1
1
1
65
79
84
87
80
70
83
85
40
82
1
1
1
1
1
1
1
1
1
1
Table 1) with varying grid search parameters. The
results of this exercise (Table 2) show that the solution
percentage is a stable measure over a relatively wide
range of grid search parameters. A statistical assessment
of the results indicated that the correlation coefficients
(i.e. for the number of solutions against the number of
tested tensors) are strongly significant at the 99%
significance levels. See Table 2 for the presentation of
the results of the tests. This statistical correlation
between the number of solutions and the number of
trial tensors is inferred to indicate that the solution
percentage is a stable measure of the inversion results.
As a further improvement to constraining the value of
the solution percentage we suggest that the results from
such an exercise (i.e. varying grid search parameters)
could be subjected to a regression analysis with the slope
of the resulting regression line indicated as a ‘Global’
solution percentage.
a
Columns A, B and C are fault plane dip azimuth, fault plane
dip and fault-slip sense (normal=1, reverse=1).
7. Examples of using DIPSLIP.BAS with published data
Table 2
Results of experiment to analyse effects of varying grid search
parameters (i.e. number of tested tensors) on solution
percentage for two test datasetsa
Solutions
Dataset A
1179533
588010
12600
6267
495
241
Correlation coefficient
Regression slope
Dataset B
75038
49234
598
355
26
16
Correlation coefficient
Regression slope
Tested tensors
7484040
3742020
61680
30840
2450
1225
0.9999961
0.1573717
7484040
3742020
61680
30840
2450
1225
0.9903395
0.0104719
Solution
percentage
0.157606453
0.157137054
0.204280156
0.203210117
0.202040816
0.196734694
0.010026403
0.013157065
0.009695201
0.011511025
0.010612245
0.013061224
a
Correlation coefficients (that attempt to indicate level of
association between number of solutions obtained and number
of tested tensors) for Table A and B, respectively, are 0.9999961
and 0.9903395. Best-fit regression lines describing a linear
relationship between number of obtained solutions and number
of tested tensors produced slopes of 0.1573717 and 0.0104719
for datasets A and B, respectively. See text for further
discussion.
The program has been extensively tested with a
variety of synthetic and real data. This section describes
the results of two of such experiments that aim to
indicate the validity of the program approach and the
resulting inversion results using previously published
data.
7.1. Example 1
The use of the program is illustrated by analysing the
fault input data file CANGEX2.TXT in Table 3,
Table 3
Fault slip data presented by Bellier et al. (1989) forming input
file CANGEX2. TXTa
A
B
C
D
E
037
042
234
203
198
060
016
208
194
242
160
245
44
40
50
70
64
48
45
56
47
60
42
85
037
030
158
148
199
032
035
176
148
199
162
160
44
39
16
56
64
44
44
52
36
52
42
40
1
1
1
1
1
1
1
1
1
1
1
1
a
Columns A and C are fault plane dip azimuths and lineation
plunge azimuths respectively, B is fault plane dip, C is lineation
plunge and E is sense of displacement on fault plane (with
normal faults=1, reverse faults 1).
779
T. Orife et al. / Computers & Geosciences 28 (2002) 775–781
originally presented in Bellier et al. (1989). Table 3 is a
listing of the original stereographic data presented in
their Fig. 11; site 10.3. The data were analysed using the
INVERS stress inversion program of Sperner et al.
Table 4
Comparison of results of fault slip inversion analysis for input
data file CANGEX2.TXT a
Stress analysis
s1
s2
s3
f
Bellier et al.
(1989)
Program
INVERS
098–64
287–26
195–04 0.46
097–71
277–19
007–00 0.429
a
Using program INVERS with results obtained for original
data by Bellier et al. (1989). Bellier et al. (1989) quote stress
ratio results as R. Note that f ¼ 1 R: See text for further
discussion.
N
(a)
σ3 stress axes
σ1 stress axes
(c)
(b)
N
970
947
950
Frequency
930
of
compatible 910
solutions
949
932
932
885
890
870
0
0.2
0.4
0.6
0.8
Stress Ratio
Fig. 3. Example of results of DIPSLIP.BAS. Palaeostress
tensors determined to be compatible with input data file
CANGEX2.TXT. Stereograms are lower hemisphere, equalarea projections with arrows indicating location of highest
density contour. Stars indicate orientation of principal stress
axes determined by Bellier et al. (1989). See text for discussion.
(a) s1 principal stress axes. Contours are 3%, 5% and 7% per
1% area. (b) s3 principal stress axes. Contours are 2%, 3% and
4% per 1% area. Obtained solutions give s3 stress axes in wide
variety of orientations, though modal solutions (indicated by
the density contours) have near horizontal plunge. (c)
Histogram of stress ratios of stress tensors.
(1993). A comparison (see Table 4) of the INVERS
results with those of Bellier et al. (1989) (see their Table 3)
show the stress axes and magnitudes of both analyses to
be very similar.
The data were then analysed using the program
DIPSLIP.BAS (though the program does not utilise any
of the lineation information presented in Table 3).
During running of DIPSLIP.BAS, the grid search
parameter of 51 was used, with increments of 0.2 for
f: These settings determine the thoroughness of the
search, i.e. the total number of trial stress tensors used.
Of the total 30,840 palaeostress tensors considered, only
4645 were found to explain the slip senses of all 12
faults.
The results of this analysis are presented in Fig. 3.
Fig. 3A and B are the density patterns of compatible s1
and s3 axes, respectively. The respective centres of these
density patterns are interpreted to broadly centre about
the orientation of the modal or ‘most-likely’ stress axes.
These results indicate that the program correctly
identifies the range of feasible s1 and s3 stress axes
(indicated in the density patterns of Fig. 3A and B)
predicted by both program INVERS and the Bellier et al.
(1989) analyses. The f values presented in Fig. 3C for
this analyses are highest at 0.4 (compared with 0.429 and
0.46 indicated by program INVERS and Bellier et al.,
Table 5
Data from sheared dykes forming input file (DAP. TXT)a
A
B
C
186
208
127
125
096
076
053
045
025
002
354
344
337
335
330
324
294
289
297
304
314
333
75
9
70
84
88
80
60
52
70
74
76
75
71
62
35
74
50
78
76
84
84
86
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
a
Columns A, B and C are fault plane dip azimuth, fault plane
dip and fault slip sense (normal=1, reverse=1).
780
T. Orife et al. / Computers & Geosciences 28 (2002) 775–781
explain the shear senses of all 22 dykes. The principal
stress orientations and f values of these tensors are
shown in Fig. 4.
Both examples illustrate the fact that sense of slip data
is often unable to identify a single stress tensor, but
instead find a range of compatible stress orientations
and f ratios.
N
(a)
σ3
σ2
σ1
(b)
50
8. Conclusion
44
40
Frequency of
compatible 30
solutions 20
22
10
10
0
38
DIPSLIP.BAS allows the estimation of palaeostresses
from data that would be inadequate for analysis using
the standard methods of fault-slip analysis. Sense of
movement data clearly contains less information than
the data on the directions of fault slip. For this reason,
fault-sense analysis is less efficient at defining the
palaeostress tensor than fault-slip analysis. Nevertheless,
fault-sense analysis broadens the scope of palaeostress
analysis allowing it to be undertaken in wider sets of
circumstances.
Acknowledgements
0
0
0.2
0.4
0.6
0.8
Stress Ratio
Fig. 4. Example of results of DIPSLIP.BAS. Palaeostress
tensors found to be compatible with data of Davidson and
Park (1978). (a) Lower hemisphere, equal-area stereogram of
principal stress axes, (b) Histogram of stress ratios of stress
tensors.
1989, respectively) though there seems be no clear trend
indicated by the histogram.
7.2. Example 2
Another illustration of the use of the program is an
analysis of data presented by Davidson and Park (1978).
Their data come not from faults but from sheared dykes.
The internal foliation within the intrusions indicate that
margin parallel displacements have occurred, and the
oblique foliation alignment in individual dykes on
vertical outcrop surfaces allows the shear sense to
determined as normal or reverse. The dykes are therefore assumed to provide equivalent dynamic information to faults of known dip-slip sense. The input file
containing these data (DAP.TXT) is shown in Tables 4
and 5. During running of DIPSLIP.BAS for this
example, the same grid search parameters and increments as in example 1 were used and of the total 30,840
palaeostress tensors considered, only 114 were found to
Discussions with Norman Fry and Chris MacLeod
are acknowledged. Mark Jessell and an anonymous
reviewer are thanked for their useful comments. Tobore
Orife is supported by Amerada Hess Ltd., Shell
International Exploration and Production (SIEP-RTS)
and the Committee of Vice-Chancellors and Principals
of the Universities of the United Kingdom (CVCP)
Overseas Scholarship (ORS) award ORS/98047010. Luis
! y
Arlegui is funded by the Ministerio de Educacion
Cultura [Programa FPI Extranjero].
References
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Formation, with Applications to Britain. Oliver & Boyd,
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Angelier, J., 1994. Fault slip analysis and palaeostress
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Deformation. Pergamon Press, Oxford, pp. 53–100.
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