Focal Mechanisms Managing, Classification And Plot Program. The FMC Manual
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FMC
A program to manage, classify and plot focal
mechanism data
version 1.3, October 2018
José A. Álvarez-Gómez
Faculty of Geology. Universidad Complutense de Madrid
This text is distributed under the Creative Commons License:
Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
1
Contents
I
The manual
4
1
Introduction
4
2
Versions
4
3
Installation
5
4
Usage
5
4.1
Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
4.2
output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.3
Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4.4
Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
5
II
License
19
The background
20
6
Introduction
20
7
The focal mechanism
21
8
Diagram
23
9
Focal Mechanism Classification
26
9.1
Comparison between rake-based and SMT axes-based classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
28
31
2
Acknowledgments
I have been using different versions of this program during the last decade.
Initially I reworked some of the Gasperini and Vannucci (2003) FORTRAN
subroutines on Matlab in order to obtain all the focal mechanism parameters from the Harvard CMT psmeca formatted catalog. During my research
on seismotectonics I started to use the Frohlich and Apperson (1992) diagram, but after Kagan (2005) I decided to try the Kaverina et al. (1996) one.
From the original Matlab program I jumped to the Free Software world
adapting it to Octave. The program only produced the x and y positions of
the events and all the plotting was done by means of GMT (Wessel et al.,
2013).
Some colleagues wanted to use the diagram for their work, but they were
not familiar with GMT, so I decided to make a big improvement in the
program to make it easy to use, distributable and with plotting support. I
choose to program it in Python with the following basic ideas:
1. It should be called from the terminal or the command line in order to
be incorporated into shell scripts
2. It should behave like any other shell unix tool, compatible with redirection, piping and ASCII format
3. it should be compatible with the GMT psmeca formats to allow the
mapping of the focal mechanisms
4. It should has the option to produce a Kaverina et al. (1996) type classification diagram
Part of the programming was done during my happy days as Ph.D. candidate at the UCM; so I have to acknowledge the UCM scholarship that
allowed me to start my scientific career.
I would like to thank the beta testers Jorge L. Giner-Robles and Alberto
Jiménez-Díaz for their comments and suggestions. Dr. Andrei Bala (National Institute for Earth Physics, Romania) has tested different versions
and suggested several improvements.
If you use FMC, and consider it appropriate, you can acknowledge it by
citing the following reference:
• Álvarez-Gómez, J.A. (2014) FMC: a one-liner Python program to manage, classify and plot focal mechanisms. Geophysical Research Abstracts, Vol. 16, EGU2014-10887.
3
Part I
The manual
1
Introduction
This is the user manual for the program FMC (from Focal Mechanisms Classification). This program was originally developed on Python 2.7.3 and
adapts some of the Gasperini and Vannucci (2003) FORTRAN routines to
obtain the different parameters of the earthquake focal mechanisms depending on the input format. Since version 1.3 it is compatible with Python
3. Several output formats are eligible, clustering analysis of the data can be
performed and a classification diagram for the focal mechanisms is produced. The default input and output formats are the same used by the
GMT program psmeca in order to make the program integration easiest
and facilitate the mapping of the data.
The program basically takes the focal mechanisms data, computes the different parameters that can be obtained, classifies each focal mechanism in
one of seven possible types, perform a clustering analysis of the data if it
is required by the user and optionally outputs the parameters in different
formats and generates a classification diagram from the input data.
2
Versions
1.3
New features:
• Added support for Pyhton 3
• Added new base diagram labels and tick marks.
• Added symbol labeling.
1.2
New features:
• Added slip sense and plunge parameters for each focal mechanism
nodal plane.
1.1
New features:
• New custom output parsing. The user can choose between the standard predefined output or any parameters in any order.
4
• Added a header with the parameter name in the output.
• Added symbol coloring.
• Added hierarchical clustering with several methods and options.
1.01 New features:
• Column number is now included in the header of the output file with
all the parameters.
Bug fixes:
• Error in the nd2ar function to obtain the tensor components from the
nodal planes.
1.0 Initial release of FMC with basic plotting function and focal mechanims
management.
3
Installation
FMC has been programmed in Python 2.7.3 and uses several common python
libraries: sys, argparse, os, NumPy and matplotlib. Depending on your
operative system and Python installation you could need to install some of
the libraries manually.
Since version 1.3 FMC works also on Python 3. In order to work properly
FMC needs NumPy version 1.14 or higher.
For Windows and Mac users a good starting point is the installation of predefined packages. From www.SciPy.org you can download program packages that includes Python, NumPy and matplotlib, as well as other modules
and IDEs. With one of these packages installed FMC should work properly.
For Linux users all the required software can be easily installed through
your favorite distribution repository (the python modules sys, argparse
and os are usually installed with the default python installation).
4
Usage
The user interface is as simple as a “command line”, or a “terminal”. I decided to keep the program as simple as possible so it can be integrated into
5
scripts with other programs such as GMT, AWK (or gawk) or any other tool
running from a shell or script (in *nix) or a command window or batch file
(in Windows/DOS). In the following the commands are written assuming
that your python installation recognizes automatically the python format.
In some systems, depending on your personal configuration, you will need
to call first Python (like “python FMC.py...” or ”python3 FMC.py...”) or
in a *nix system with the “./” needed to execute a script (like ./FMC.py
...).
In *nix you will need to give execution permission to the program:
chmod +x FMC.py
To run FMC on a file simply type in the terminal:
FMC.py input-file.dat
By default FMC will read the input file as a psmeca Centroid Moment Tensor (CMT) format (in Harvard convention). This kind of file can be downloaded directly from Global CMT. The output will be shown on the screen
in psmeca CMT format so it can be directly pipe into psmeca:
FMC.py input-file.dat | psmeca -R {...etc}
Or stored as an ASCII file:
FMC.py input-file.dat > output-file.dat
In this case FMC will add a new column at the end of each register with the
focal mechanism type.
Alternatively the input file can be a single plane in Aki and Richards (1980)
convention (Right Hand Rule for the plane orientation, and rakes from 0
to 180 in reverse faults and 0 to -180 in normal faults, 0 for left-lateral and
±180 for right-lateral). This format is also compatible with psmeca.
4.1
Input
FMC input can be given as an ASCII file or as standard input, from a pipe
(“|”) or a redirection (“<”). The following codes are then equivalent:
FMC.py input-file.dat
cat input-file.dat | FMC.py
FMC.py < input-file.dat
If no input is given, then FMC will show the on-screen help.
6
The input format is specified with the optional flag “-i”, and the possible
values are:
CMT Harvard Centroid Moment Tensor (by default):
[longitude, latitude, depth, mrr, mtt, mff, mrt, mrf, mtf, Exponent
(dyn-cm), X plot, Y plot (for GMT), ID]
AR Aki and Richards one plane convention:
[longitude, latitude, depth, strike, dip, rake, magnitude (Mw),
X plot, Y plot (for GMT), ID]
4.2
output
FMC output format can be selected among the following options with the
flag “-o”:
CMT Harvard Centroid Moment Tensor (psmeca compatible):
[longitude, latitude, depth, mrr, mtt, mff, mrt, mrf, mtf, Exponent
(dyn·cm), X plot, Y plot (for GMT), ID, TYPE]
P Focal mechanism both nodal planes (psmeca compatible):
[longitude, latitude, depth, strike A, dip A, rake A, strike
B, dip B, rake B, Scalar seismic moment mantissa, Exponent (dyn·cm),
X plot, Y plot (for GMT), ID, TYPE]
AR Focal mechanism one plane (psmeca compatible):
[longitude, latitude, depth, strike, dip, rake, magnitude (Mw),
X plot, Y plot (for GMT), ID, TYPE]
K Kaverina diagram position for plotting outside FMC:
[X Kaverina diagram, Y Kaverina diagram, Mw, Depth, ID, TYPE]
ALL All the parameters obtained:
[longitude, latitude, depth, mrr, mtt, mff, mrt, mrf, mtf, Exponent
(dyn·cm), Scalar seismic moment (dyn·cm), Mw, strike A, dip
A, rake A, strike B, dip B, rake B, Slip trend A, Slip plunge
A, Slip trend B, Slip plunge B, P trend, P plunge, B trend,
B plunge, T trend, T plunge, fclvd, X Kaverina diagram, Y Kaverina
diagram, ID, TYPE]
CUSTOM In case you need any focal mechanism parameters in any order you can use the CUSTOM option and give the requested parameters in any order using the flag “-of”. The output parameters
need to be listed separated by commas. The accepted parameters
names are listed below, and can be seen on the terminal using FMC.py
-helpFields
7
lon longitude
lat latitude
dep depth
mrr mrr centroid moment tensor component
mtt mtt centroid moment tensor component
mff mff centroid moment tensor component
mrt mrt centroid moment tensor component
mrf mrf centroid moment tensor component
mtf mtf centroid moment tensor component
mant mantissa of the seismic moment tensor
expo exponent of the seismic moment tensor
Mo Scalar seismic moment
Mw Moment (or Kanamori) magnitude
strA Strike of nodal plane A
dipA Dip of nodal plane A
rakeA Rake of nodal plane A
strB Strike of nodal plane B
dipB Dip of nodal plane B
rakeB Rake of nodal plane B
slipA Slip sense of plane A
plungA Plunge of slip vector of plane A
slipB Slip sense of plane B
plungB Plunge of slip vector of plane B
trendp Trend of P axis
plungp Plunge of P axis
trendb Trend of B axis
plungb Plunge of B axis
trendt Trend of T axis
plungt Plunge of T axis
fclvd Compensated linear vector dipole ratio
x_kav x position on the Kaverina diagram
y_kav y position on the Kaverina diagram
ID ID of the event
clas Focal mechanism rupture type
posX X plotting position for GMT psmeca
posY Y plotting position for GMT psmeca
8
Examples of use
Obtaining nodal planes from moment tensor
Command:
echo -2.54 37.09 12 -3.4669 -2.0652 5.5321 6.2368 -1.8004 -5.1775 22 X Y ID | FMC.py -o P
Result:
#Longitude Latitude Depth_(km) Strike_A Dip_A Rake_A Strike_B Dip_B Rake_B Seismic_moment_mantissa
Exponent_(dyn-cm) X_position(GMT) Y_position(GMT) ID Rupture_type
-2.54 37.09 12.0 190.925 42.4899 -20.9735 296.709 76.0089 -130.541 9.6045 22.0 X Y ID N-SS
Obtaining moment tensor from one nodal plane
Command:
echo -2.54 37.09 12 190.925 42.4899 -20.9735 4.6 X Y ID | FMC.py -i AR -o CMT
Result:
#Longitude Latitude Depth_(km) mrr mtt mff mrt mrf mtf Exponent_(dyn-cm) X_position(GMT) Y_position(GMT) ID
Rupture_type
-2.54 37.09 12.0 -3.56563 -2.21928 5.78491 6.70126 -1.61249 -5.19047 22.0 X Y ID N-SS
Obtaining all the parameters from the moment tensor
Command:
echo -2.54 37.09 12 -3.4669 -2.0652 5.5321 6.2368 -1.8004 -5.1775 22 X Y ID | FMC.py -o ALL
Result:
#Longitude Latitude Depth_(km) mrr mtt mff mrt mrf mtf Exponent_(dyn-cm) Seismic_moment_Mo Magnitude_Mw Strike_A
Dip_A Rake_A Strike_B Dip_B Rake_B Slip_trend_A Slip_plunge_A Slip_trend_B Slip_plunge_B Trend_P Plunge_P
Trend_B Plunge_B Trend_T Plunge_T fclvd X_Kaverina Y_Kaverina ID Rupture_type
-2.54 37.09 12.0 -3.4669 -2.0652 5.5321 6.2368 -1.8004 -5.1775 22.0 9.6045e+22 4.6 190.925 42.4899 -20.9735
296.709 76.0089 -130.541 206.709 -13.9911 100.925 -47.5101 167.141 43.8185 308.393 39.1024 56.0979 20.5155
0.0445259 -0.243839 0.0899979 ID N-SS
Obtaining all the parameters from CMT input file and storing to an ASCII
file
Command:
FMC.py -o ALL japan_CMT.dat > Japan_parameters.dat
Using CUSTOM output to obtain event location and slip vector of both
nodal planes
Command:
echo -2.54 37.09 12 -3.4669 -2.0652 5.5321 6.2368 -1.8004 -5.1775 22 X Y ID | FMC.py -o CUSTOM -of
lon,lat,slipA,plungA,slipB,plungB
Result:
#Longitude Latitude Slip_trend_A Slip_plunge_A Slip_trend_B Slip_plunge_B
2.54 37.09 206.709 -13.9911 100.925 -47.5101
9
4.3
Plot
Optionally FMC will produce a classification diagram. FMC uses matplotlib
libraries and can generate figures in different formats (emf, eps, jpeg, jpg,
pdf, png, ps, raw, rgba, svg, svgz, tif, tiff). The format is determined automatically from the plot file name extension.
The diagram uses the Kaverina et al. (1996) projection technique, used also
by Kagan (2005); but incorporating a classification similar to the geological conceptual classification of faults. The earthquakes are classified into
seven types according to the values of the P, T and B Centroid Moment
Tensor axes following a simple algorithm (Figure 4.1), and are represented
conveniently on the Kaverina diagram (Figure 4.2). This classification is
very similar to the used by Johnston et al. (1994).
10
Figure 4.1: Focal mechanism classification algorithm flow chart.
11
Figure 4.2: Classification diagram. N: Normal; N-SS: Normal - Strike-slip;
SS-N: Strike-slip - Normal; SS: Strike-slip; SS-R: Strike-slip - Reverse; R-SS:
Reverse - Strike-slip; R: Reverse.
FMC uses several flags in order to customize the plot.
-p This flag activates the plotting. It must be followed by the name of the
figure file that will be produced. The name used for the file (without
the extension) is used as title for the plot.
-pc With this flag the user specifies the parameter that is used to fill the
symbols. A color palette is produced with the range of the selected
parameter values.
-a This flag is used to annotate the symbols with a certain parameter.
With -pc and -a the parameters must be given with its corresponding internal name as listed in section 4.2 on page 7.
Examples of use
Plotting data from standard input
Command:
echo -2.54 37.09 12 -3.4669 -2.0652 5.5321 6.2368 -1.8004 -5.1775 22 X Y ID | FMC.py -p My data.png
12
Figure 4.3: Plot result from the command echo -2.54 37.09 12 -3.4669
-2.0652 5.5321 6.2368 -1.8004 -5.1775 22 X Y ID | FMC.py -p My
data.png
Plotting data from input file and shading the symbols with a parameter
Command:
FMC.py -p 'Japan 2011 data.png' japan_CMT_2011.dat -pc fclvd
Figure 4.4: Plot result from the command FMC.py -p 'Japan 2011
data.png' japan_CMT_2011.dat -pc fclvd
Plotting data from input file shading and annotating the symbols
Command:
FMC.py Japan_CMT_2011_biggest.dat -p "Japan 2011 biggest events.png" -pc dep -a Mw
13
Figure
4.5:
Plot
result
from
the
command
FMC.py
Japan_CMT_2011_biggest.dat -p "Japan 2011 biggest events.png"
-pc dep -a Mw
Plotting data from input file and piping to psmeca
Command:
FMC.py -p 'Japan data.pdf' japan_CMT.dat | psmeca -R125/170/30/60 -JM8c -Ba10f2WESN -Sm5p/-1 > CMT_map.ps
130˚
140˚
150˚
160˚
170˚
60˚
60˚
50˚
50˚
40˚
40˚
30˚
30˚
130˚
140˚
150˚
160˚
170˚
Figure 4.6: Map generated with psmeca (GMT) from the FMC output.
14
4.4
Clustering
FMC implements the hierarchical agglomerative clustering algorithms from
SciPy (scipy.cluster.hierarchy) in order to group the data1 . The advantages
of this algorithms are its versatility, as the user can choose between a number of metrics and grouping methods, its capacity to automatically select a
minimum number of clusters without an a priori estimation, and its potential to work with different parameters with different scales and with strong
different populations in clusters.
The parameters for the clustering are passed by several optional flags. If
any of the following flags is given in the command line FMC will perform the
clustering analysis using some default options if needed. When a clustering
analysis is done, by default FMC will shade the symbols in the diagram using
the cluster number, unless a different parameter is stated with -pc flag.
-cm Method to be used in the clustering analysis. The options are:
single single/min/nearest
d(u, v) = min(dist(u [i ] , v [ j]))
complete complete/max/farthest point
d(u, v) = max(dist(u [i ] , v [ j]))
average average/UPGMA
d(u, v) =
∑
ij
d(u [i ] , v [ j])
(|u| ∗ |v|)
weighted weighted/WPGMA
d(u, v) = (dist(s, v) + dist(t, v))/2
centroid centroid/UPGMC [default]
dist(s, t) = kcs − ct k2
where cs and ct are the centroids of clusters s and t, respectively.
When two clusters s and t are combined into a new cluster u, the
new centroid is computed over all the original objects in clusters
s and t. The distance then becomes the Euclidean distance between the centroid of u and the centroid of a remaining cluster v
in the forest.
1 The
details on the clustering algorithms shown in this section are taken from the SciPy
documentation.
15
median median/WPGMC, assigns d(s, t) like the centroid method.
When two clusters s and t are combined into a new cluster u, the
average of centroids s and t give the new centroid.
ward Ward variance minimization algorithm.
r
|v| + |s|
|v| + |t|
|v|
d(u, v) =
d(v, s)2 +
d(v, t)2 −
d(s, t)2
T
T
T
where u is the newly joined cluster consisting of clusters s and t,
v is an unused cluster in the forest,T = |v| + |s| + |t|, and |∗| is
the cardinality of its argument.
Methods ”centroid”, ”median” and ”ward” are correctly defined only
if Euclidean pairwise metric is used.
-ce Metric used to measure distances between events parameters. These
metrics work with non-Boolean vectors. By default FMC uses euclidean
distance.
braycurtis The Bray-Curtis distance between two points u and v is
d(u, v) =
∑i | ui − vi |
∑i | ui + vi |
canberra The Canberra distance between two points u and v is
d(u, v) =
| ui − vi |
∑ | ui | + | vi |
i
chebyshev The Chebyshev distance between two n-vectors u and v
is the maximum norm-1 distance between their respective elements. More precisely, the distance is given by
d(u, v) = max |ui − vi |
i
cityblock City block or Manhattan distance between the points.
correlation Correlation distance between vectors u and v. This is
1−
(u − u¯)·(v − v¯)
k(u − ū)k2 k(v − v̄)k2
cosine Cosine distance between vectors u and v,
1−
16
u·v
k u k2 k v k2
euclidean Distance between m points using Euclidean distance (2norm). [Default]
hamming Normalized Hamming distance, or the proportion of those
vector elements between two n-vectors u and v which disagree.
jaccard Jaccard distance between the points. Given two vectors, u
and v, the Jaccard distance is the proportion of those elements
u[i ] and v[i ] that disagree.
mahalanobis The Mahalanobis distance between two points u and v
is
q
(u − v)(1/V )(u − v)T
where 1/V is the inverse covariance matrix.
minkowski Distances using the Minkowski distance ku − vk p (p-norm)
where p ≥ 1.
seuclidean Standardized Euclidean distance. The standardized Euclidean distance between two n-vectors u and v is
q
∑ ( u i − v i )2 ÷ V [ x i ]
V is the variance vector; V [i ] is the variance computed over all
the i’th components of the points. It is automatically computed.
sqeuclidean Squared Euclidean distance ku − vk22 between the vectors.
-cn A priori number of clusters to obtain. If zero or non-present the number of clusters is automatically computed using the Elbow method.
This method uses the percentage of variance explained as a function
of the number of clusters; when the increment of number of clusters
does not improves the percentage of variance explained the number
of clusters is set.
-ci Parameters used to perform the cluster analysis. By default FMC uses
the position on the Kaverina diagram, which is a proxy for the principal moment tensor axes plunges. If the parameters given are not
in the same physical magnitude and unit the euclidean distance is
not appropriate and a different metric should be used. In these cases
the Mahalanobis distance is a good choice, as is equivalent to the Euclidean distance in the transformed space, using the covariance matrix of each parameter.
The parameters must be given with its corresponding internal name
as listed in section 4.2 on page 7.
17
Examples of use
Automatic clustering using the position in the Kaverina diagram (default)
FMC.py -p 'Japan 2011 clusters.png' japan_CMT_2011.dat -cn 0
Figure 4.7: Plot result from the command FMC.py -p 'Japan 2011
clusters.png' japan_CMT_2011.dat -cn 0
Clustering using the epicentral location
FMC.py japan_CMT_2011.dat -cn 4 -ci lon,lat
Figure 4.8: Plotting with GMT psmeca of the clusters obtained with
the
command
FMC.py -p 'Japan 2011 spatial clusters.png'
japan_CMT_2011.dat -cn 4 -ci lon,lat
Clustering using Mahalanobis metric and complete grouping with the
slip and plunge of the slip vector of nodal plane A
18
FMC.py -p 'Japan 2011 clusters plane A.png' japan_CMT_2011.dat -cn 4 -ci slipA,plungA -cm complete -ce
mahalanobis
Figure 4.9: Plotting with GMT psmeca of the clusters obtained with
the
command
FMC.py -p 'Japan 2011 clusters plane A.png'
japan_CMT_2011.dat -cn 4 -ci slipA,plungA -cm complete -ce
mahalanobis
5
License
FMC has been developed with Free Software and/or Open Source tools. FMC
uses NumPy and matplotlib which are distributed under BSD license.
FMC is distributed under GNU Generic Public License.
19
Part II
The background
6
Introduction
Seismicity is frequently used to deduce the tectonics of a region. The study
of earthquakes as a tectonic component, the seismotectonics (Scholz, 2002),
has grown as one of the key research areas on active tectonics, specially
from the analysis of earthquake focal mechanisms.
The first focal mechanism determination methods, based on P wave’s first
motion polarities, were developed on the first half of the 20th century, specially in Japan where a dense seismic network were available (Byerly, 1928,
1938; Koning, 1941; Honda and Masatuke, 1952; Scheidegger, 1957). Since
1960 the development of the digital computer allow the numerical determination of fault-plane solutions with different methods (e.g. Knopoff, 1961;
Langston and Helmberger, 1975; Dziewonski et al., 1981).
In parallel with the development of the modern seismology, the theory of
Plate Tectonics changed the way geologists understand the Earth. Since
the end of the 16th century naturalists made observations on the potential
continental drifting; but it was in the early 20th century when the theory
started to be formally proposed (Taylor, 1910; Wegener, 1912; Schwinner,
1919; Holmes, 1931). The relevant information acquired on the magnetic
stripes on the ocean floor in the 60’s (e.g. Hess, 1962; Vine, 1966; Le Pichon,
1968; Morgan, 1968), in addition to an increasing number of geological and
geophysical observations, established the Plate Tectonics as the global geodynamical paradigm on Earth Sciences. The study of earthquakes related
to Plate Tectonics was developed at the same time establishing the basic
concepts of seismotectonics and the lithospheric deformation (e.g. Benioff,
1954; Sykes, 1967; Isacks et al., 1968; Isacks and Molnar, 1969).
Since the 70’s focal mechanisms are computed in a systematic way and
global catalogs of focal mechanisms are available since then. As the amount
of data increases, our knowledge on the active tectonics improves. As a
consequence of the continuous increase of data available we need new tools
to analyze it systematically.
In order to represent focal mechanism populations Frohlich and Apperson
(1992) proposed a diagram to visualize focal mechanism data as a function
of the rupture type. This kind of representation is popular and widely used
on seismotectonics to represent the focal mechanism types of the study
areas (e.g. Frohlich et al., 1997; Borges et al., 2001; Igarashi et al., 2001;
Ratchkovski and Hansen, 2002; Kita et al., 2006; Serpelloni et al., 2007). This
20
representation is not accurate, presenting distortion problems (Frohlich,
2001), and Kagan (2005) used the Kaverina et al. (1996) projection to avoid
them. The latter projection is used in FMC by default.
7
The focal mechanism
The term “focal mechanism” is used to refer to the parameters that characterize an earthquake rupture. A focal mechanism usually presents the
characteristics of the two orthogonal possible rupture planes: strike, dip
and rake of the slip vector over the plane. The energy released by the event,
its hypocentral location and the exact time of occurrence are also given.
The focal mechanism can, instead of describe the geometry of the possible rupture planes (6 variables), describe the characteristics of the rupture
by means of the seismic moment tensor (SMT) or centroid moment tensor
(CMT) (6 variables).
The SMT is based in the representation of the equivalent forces on a point
seismic source. In this tensor the sum of angular moments should be null,
in other words, the SMT is a symmetric tensor on a three dimensional space
(9 components) with 6 independent components:
MXX
M = MYX
MZX
MXY
MYY
MZY
MXZ
MYZ
MZZ
(7.1)
where
MXY = MYX
MXZ = MZX
(7.2)
MYZ = MZY
Similarly to the strain tensor where we can define the orientation of the
principal axes (eigenvectors) and its magnitudes (eigenvalues), the SMT
can be defined by the orientation of its principal axes P, B and T (eigenvectors) and its magnitudes (eigenvalues). The T axis describes the greatest of
the eigenvalues, the P describes the lowest and the B the intermediate.
If the seismic event is generated by the slip in a fault rupture, its characteristics should be well described as a double couple moment tensor (Figure
7.1):
M0
0
0
M = 0 − M0 0
(7.3)
0
0
0
21
Figure 7.1: Double couple deformation mechanism. a) Plan view of horizontal displacement on an idealized point vertical fault A-A’ or F-F’ and resulting distribution of compressions (+) and dilatations (-). b) Focal mechanism representation on a stereographic projection. Compression quadrants
are filled. c) Horizontal displacement produced by a point size vertical dislocation computed with the Okada (1992) equations; compressional quadrants are filled while dilatational are kept empty following the focal mechanism representation. a) and b) modified from Bullen and Bolt (1985).
where M0 is the scalar seismic moment.
The most used representation of the double couple is based on the stereographic projection. In this projection both nodal planes of the focal mechanism are plotted, and the quadrants containing the T axis, the compression
quadrants, are filled (Figure 7.1). This kind of representation - commonly
called “beach-ball” - located on a map on the earthquake epicenter is useful to interpret the seismic data from a tectonic point of view, allowing the
study of the earthquake and its rupture fault plane on its tectonic context,
being one of the basis of the seismotectonics. The compressional and dilatational quadrants are separated by two orthogonal planes, the nodal planes,
being one of them the plane of rupture that generated the earthquake (Figure 7.2).
As we increase the number of focal mechanisms to study, we increase the
difficulty to interpret the data, obscuring frequently the details of a tectonic
environment if we just project them over a map. In order to facilitate the
study of groups of focal mechanisms several plots can be proposed, great
part of them based on classical structural geology plots.
The plots inherited from the structural geology are based in the geometrical
characteristics of the nodal planes of the SMT, equivalent to the characteristics of the fault planes: strike (ϕ), dip (δ) and rake (λ) (Figure 7.2). The
main problem of these representations is the duplication of data. Each focal mechanism presents two equivalent and orthogonal nodal planes. The
selection of one of them as the fault plane is not straightforward and although from a mechanical point of view they can be discriminated (McKenzie, 1969; Michael, 1987; Gephart, 1990), when both nodal planes are me22
Figure 7.2: a) Focal sphere diagram on stereographic projection on the
lower hemisphere. Nodal planes A and B, A representing the fault plane
and B the auxiliary plane, with its orientation characteristics: ϕ, strike; δ,
dip and λ, rake. The orientation of the SMT principal axes are also shown:
T, B and P. Modified from Udías (1989). b) Block diagram showing the parameters that define the orientation of a fault plane in the space: ϕ, strike;
δ, dip and λ, rake.
chanically compatible or reactivated structures can be present, additional
geological and/or geophysical information is required. There are as many
representations of this kind as possible combinations between these parameters and its derivatives (usually trigonometrical functions). These relations can be represented in biaxial plots or stereographic projections (e.g.
Davis and Reynolds, 1996; Ramsay and Huber, 1997; Pollard and Fletcher,
2005).
The focal mechanisms can also be plotted as function of the principal axes
T, B and P. These axes form an orthogonal system. The orientation of each
axis can be defined in the space as any line, by its direction and plunge. The
main advantages of these representations are: the univocality, each focal
mechanism is represented by one point; its simplicity of interpretation, the
kind of deformation (normal, reverse, strike-slip) is function of the plunge
of the axes; and the reduction of errors compared with the planes, as they
are derived from the principal axes of the SMT (Vannucci and Gasperini,
2003).
8
Diagram
Frohlich and Apperson (1992) developed a ternary diagram to represent
focal mechanisms based in the trigonometrical relation:
sin2 ι T + sin2 ι B + sin2 ι P = 1,
23
(8.1)
where ι T , ι B and ι P are the plunges of the axis T, B and P of the SMT. This
relation is true for three orthogonal axes.
As the equation of a sphere with radius unity is
x 2 + y2 + z2 = 1
(8.2)
and all the plunge angles are positive between 0°and 90°, then the representation of a focal mechanism defined by these axes is equivalent to the
projection of a point in an spherical octant over a planar surface. As a
sphere is not a developable surface can not be projected onto a plane without distortion.
The projection used by Frohlich and Apperson (1992) represents this spherical octant as a triangle, which is equivalent to the gnomonic geographical
projection. In this projection we define the angles
sin ι T
ψ = arctan
− 45◦ ,
(8.3)
sin ι P
and
1
1
ι s = arcsin √ = arctan √ ≈ 35.26◦ ,
3
2
(8.4)
which is the angle of the ternary symmetry axis with the orthogonal axes
of the octant.
From these angles the x and y coordinates over the plane are given by:
cos ι B · sin ψ
sin ι s · sin ι B + cos ι s · sin ι B · cos ψ
cos ι s · sin ι B − sin ι s · cos ι B · cos ψ
y=
sin ι s · sin ι B + cos ι s · sin ι B · cos ψ
x=
(8.5)
(8.6)
The angle ι s in the projection is the angle of the plunges of the axes T, B
and P for the focal mechanism that occupies the barycentre of the ternary
diagram where x = y = 0.
As can be seen in Figure 8.1a this projection introduces remarkable distortions towards the extremes of the diagram. This distortions could difficult
the study of some groups of focal mechanisms as the author pointed out
(Frohlich, 2001). In order to avoid these distortions Kaverina et al. (1996)
proposed the use of a projection capable of maintain the proportion of
the areas equal, equivalent to the Lambert azimuthal equal-area projection
(Figure 8.1b), which is the used in structural geology to generate the stereographic equal-area Schmidt stereonet. In this case the limits of the diagram
are formed by great circles instead of straight lines forming a triangle (Figure 8.1).
24
a)
b)
Lambert Azimuthal
Gnomonic
Equal-Area
Figure 8.1: Representation of an hemisphere of the globe centred on Europe (45°N, 15°E) in the two different projections referenced in the text: a)
Gnomonic projection; b) Lambert azimuthal equal-area. In thick lines are
marked the spheric octant and some reference lines in order to compare
both projections.
If we take the sines of the plunges of the main axes ι T , ι B and ι P :
z T = sin ι T
z P = sin ι P
(8.7)
z B = sin ι B
the length of the vector that connects the center of the diagram with the
projected point is defined by
1
L = 2 sin
· arccos
2
zT + zP + zB
√
3
(8.8)
and the normalization factor is:
N=
q
2 · [(z B − z P )2 + (z B − z T )2 + (z T − z P )2 ]
(8.9)
The coordinates of the projected focal mechanisms in the plane are defined
as follows (these equations corrects the wrong equations showed in Kagan
(2005)):
25
x=
y=
9
√
3·
L
· (z P − z T )
N
L
· (2z B − z P − zT )
N
(8.10)
Focal Mechanism Classification
In order to classify the focal mechanisms Kagan (2005) divided the octant
in three areas (Figure 9.1b) corresponding to the three basic Andersonian
regimes: normal, reverse and strike-slip. The dividing lines start at the diagram center and run through the middle of each great circle (dashed lines
in Figure 9.1b). This classification, although simple and straightforward, is
too basic sometimes for a detailed seismotectonic study.
a)
Strike‐slip
b)
Strike‐slip
Odd
Normal
Normal
Reverse
Frohlich & Apperson, 1992
Reverse
Kaverina et al., 1996
Kagan, 2005
Figure 9.1: Focal mechanisms classification diagrams based on SMT axes
plunges proposed by a) Frohlich and Apperson (1992) and b) Kagan (2005).
From the seismotectonic point of view an earthquake reflects the brittle
lithospheric strain produced by tectonic processes. The vast majority of
earthquakes are produced by displacements on faults (some other processes as deep volume changes or cave collapses can produce seismic signals too), and they are interpreted usually as fault ruptures. Following
this reasoning the most appropriate classification for the focal mechanism
should be done by means of the slip vector on the fault, given by the rake
angle. This classification can be done following one of the proposed conventions (Rickard, 1972; Aki and Richards, 1980; Angelier, 1994). As has
been mentioned before, the problem is the duplicity of data due to the two
nodal planes solution of the focal mechanism.
26
Figure 9.2: Focal mechanism classification diagram. N) Normal; N-SS) Normal - Strike-slip; SS-N) Strike-slip - Normal; SS) Strike-slip; SS-R) Strike-slip
- Reverse; R-SS) Reverse - strike-slip and R) Reverse.
An extended discussion on the relation among the different focal mechanism parameters can be found on Célérier (2010); the author also points
out the difficulty on interpreting the stress state from the focal mechanisms,
specially where oblique faulting takes place and reactivation of inherited
structures is possible (Wyss et al., 1992).
However the SMT can be interpreted as a seismic strain tensor (Kostrov,
1974) where the main axes are equivalent to the SMT main axes (Wyss et al.,
1992). With this relation in mind we can interpret the populations of focal mechanisms as representations of strains in different tectonic settings.
Oblique slip focal mechanisms with normal component can be interpreted
as transtensional strain, while oblique slip focal mechanisms with reverse
component can be interpreted as transpressional.
In order to count with a classification more detailed than the one of Kagan
(2005) I decided to classify the focal mechanism in a series of fields that
include the oblique slip regimes (Álvarez-Gómez, 2009). This approximation is similar to the Johnston et al. (1994) classification; with 7 classes of
earthquakes: 1) Normal; 2) Normal - Strike-slip; 3) Strike-slip - Normal; 4)
Strike-slip; 5) Strike-slip - Reverse; 6) Reverse - Strike-slip and 7) Reverse.
The resulting diagram incorporating this 7 fields classification to the Kaverina projection is shown in Figure 9.2.
The algorithm to classify the focal mechanisms based on the SMT main axes
plunges is the shown in Figure 4.1.
When one of the three main SMT axes is greater or equal than 67.5◦ (reπ
sulted from 3 ∗ 4/2 ) we obtain a "pure" Andersonian tectonic regime. From
27
Figure 9.3: Diagram showing the convention used in this work for the rake
classification of faults or focal mechanism nodal planes. Inside the grey
circle the equivalent SMT axes-based classification is shown (Figure 8.1).
the pure normal or reverse faulting to the strike-slip faulting there is a range
of oblique faulting types with more or less relevance of the horizontal component. On the other hand between the pure normal and pure reverse faulting there is a permutation of the T and P axes in the vertical position, while
the B axis remains horizontal. The transition takes place when the P and T
axes plunge equally and a vertical nodal plane is present.
In the following section I compare the results of both classifications, the one
based in the main axes plunges of the moment tensor and the one based on
the fault rakes.
9.1
Comparison between rake-based and SMT axes-based classifications
In order to classify the nodal planes according to its rakes I adopted a convention hybrid between the Rickard (1972) and Aki and Richards (1980)
conventions. The focal mechanism rakes are usually given with the Aki
and Richards (1980) convention, while from a geological point of view the
Rickard (1972) convention is more used. I define 7 equivalent classes to
those used in the SMT axes-based classifications (Figure 9.3).
Each one of the nodal planes of the focal mechanisms from the Global CMT
catalog (Ekström et al., 2012) is classified according to the rake classification
shown in Figure 9.3. From the analysis of the focal mechanism alone we
cannot choose unequivocally the fault plane responsible of the earthquake.
To analyze the degree of uncertainty I show in Figure 9.4a the relative frequency of relations between nodal planes rupture types. For example, it
28
can be seen that when one nodal plane is classified as normal, the other can
be normal as well, or can present some strike-slip component, but it can
not present reverse component; the same can be said for the reverse nodal
planes. When one nodal plane is pure strike-slip, the other can present dipslip component too, ranging from normal with strike-slip component (NSS) to reverse with strike-slip component (R-SS), but being more frequent
the relations between nodal planes of mainly strike-slip type.
Comparing the focal mechanism classified with the SMT axes and the nodal
planes classified with the rakes (Figure 9.4b) a similar picture as the described above can be seen. The percentages of relations change sensibly,
but the common picture is the same. When the focal mechanism is classified as normal type we can expect that both nodal planes are of normal
type or normal with some strike-slip component. Similarly when the focal
mechanism is classified as reverse, both nodal planes are mainly of the reverse type too. For the case of the strike-slip focal mechanism almost all
the nodal planes are of strike-slip type too.
We can conclude that there is not much difference between the results obtained from the rake classification of the nodal planes and the classification of the focal mechanism attending to the SMT axes plunges. The focal
mechanism classification based on the SMT axes has the advantage of the
univocallity of its classification, in contrast with the duplicity of data and
uncertainty related to the nodal planes rake classification.
29
Figure 9.4: Heatmaps of the proportion of relations between a) both nodal
planes rupture type based on the rake classification, and b) between the focal mechanism classification and the nodal planes rupture types rake classification. The data used is the entire Global CMT catalog (Ekström et al.,
2012). Numbers show percentages over the entire catalogue.
30
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