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Software
Here is the software implementation of a function which
calculates the direction of maximum
horizontal stress using only the directions of the principal
stresses and the stress magnitude ratio R = (S1 - S2)/(S1 -
S3). The equation used is Equation 11 of
Lund and Townend
(2007) but with added checks on the suitability of Eq. 11.
Implementations are currently available in Python, C,
Fortran and Matlab.
If you have another favourite computer language, feel free
to reimplement the algorithm. If you do, please send me a
copy.
Implementations
The SH() function takes the three principal stress axes and
a value of R as arguments. It is important
that the three stress vectors are orthonormal, i.e. have
unit magnitude and are orthogonal to eachother. The SH()
function checks for unit magnitudes and normalizes the
vectors if they are not of unit length. It also checks for
orthogonality but does NOT
change the direction of the vectors, only reports if they
are not orthogonal. Finally the value of R is checked.
I have some benchmark tensors
against which you can check
your own code. The first three input tensors are identical
to those used in
Lund and Townend
(2007). In the file,
columns 1 and 2 have trend and plunge in degrees, and
columns 3, 4 and 5 have the corresponding coordinates in the
geographical north, east and down coordinate system. The
stress tensors are listed using two
lines, on the first is S1 and on the second S3. S2 is
calculated with a cross-product. The resulting
directions of maximum horizontal stress are
here.
If you use this code for work which is intended for
publication, please cite the
Lund and Townend
(2007) paper.
- Python
- The SH function
- A program which
takes the trend and plunge of S1
and S3, calculates S2 and then calculates the
direction of SH for a range of R-values. You also
need the library
BL.py.
- C
- The SH function
- A program which reads
a file with S1 and S3
principal stress directions, in (NED) coordinates,
and calculates the direction of SH for a range of
R-values. I have a simple
Makefile and the
example stress tensors are
here.
- Fortran
- The SH function
, works both under Fortran 77 and Fortran 90.
- A program which
reads a file with S1 and S3
principal stress directions, in (NED) coordinates,
and calculates the direction of SH for a range of
R-values. This is Fortran 90 code.
The example stress tensors are
here.
- Matlab
Updates
- 10 May 2011: There are a number of reports saying the
Matlab code works, so this is now an official release.
- 18 April 2008: There was a factor of 2 error in the
calculation of the second term in the second derivative in
all language functions.
Correcting this, from a 4 to a 2, did not change any of
the results in Lund and Townend (2007).
Thanks to Oliver Heidbach and his group for finding this.
Disclaimer
The usual stuff: This program is distributed in the hope
that it will be useful, but WITHOUT ANY WARRANTY; without
even the implied warranty of MERCHANTABILITY or FITNESS FOR
A PARTICULAR PURPOSE.
Reading material
The background for the calculation of the direction of the
maximum horizontal stress, SH, using only limited
information on the stress tensor, can be found in:
Lund B. and J. Townend,
2007, Calculating horizontal stress
orientations with full or partial knowledge of
the tectonic stress tensor, Geophys. J. Int., 170,
1328 - 1335, doi: 10.1111/j.1365-246X.2007.03468.x
This paper contains examples showing how much the direction
of SH varies with different R, and how it varies with the
plunge of the largest subhorizontal principal stress. There
is also a section in the paper showing the importance of a
correct calculation of SH using real data from the Grimsey
lineament in north Iceland.
More information on stress, stress inversion and the
direction of maximum horizontal stress can be found in my
thesis (Lund, 2000).
Other, more recent, sources for information on obtaining
the state of stress from earthquake data are, for example:
Arnold R. and J. Townend, 2007. A Bayesian approach to
estimating tectonic stress from seismological data, Geophys.
J. Int., 170, 1336-1356, doi: 10.1111/j.1365-246X.2007.03485.x
Hardebeck J. and A. Michael, 2006. Damped regional-scale
stress inversions: Methodology and examples for southern
California and the Coalinga aftershock sequence, J. Geophys.
Res., 111, B11310, doi: 10.1029/2005JB004144.
Townend, J. and M.D. Zoback, 2006. Stress, strain, and
mountain building in central Japan, J. Geophys. Res., 111,
B03411, doi:10.1029/2005JB003759.
Boness, N.L. and M.D. Zoback, 2006a. Mapping stress and
structurally controlled crustal shear velocity anisotropy
in California, Geology, 34, 825-828,
doi:10.1130/G22309.1.
Boness,N.L. and M.D. Zoback, 2006b.A multiscale study of the
mechanisms controlling shear velocity anisotropy in the San
Andreas Fault Observatory at Depth, Geophysics, 71,
F131-F146.
Hardebeck, J.L. and E. Hauksson, 2001. Stress orientations
obtained from earthquake focal mechanisms: What are
appropriate uncertainty estimates?, Bull. Seism. Soc. Am.,
91, 250-262.
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