pyrocko.moment_tensor¶
This module provides various moment tensor related utility functions.
It can be used to convert between strikediprake and moment tensor representations and provides different options to produce random moment tensors.
Moment tensors are represented by MomentTensor
instances. The
internal representation uses a northeastdown (NED) coordinate system, but it
can convert from/to the conventions used by the Global CMT catalog
(upsoutheast, USE).
If not otherwise noted, scalar moment is interpreted as the Frobenius norm
based scalar moment (see MomentTensor.scalar_moment()
. The scalar
moment according to the “standard decomposition” can be found in the
output of MomentTensor.standard_decomposition()
.
Functions

Convenience function to convert various objects to moment tensor object. 

Given euler angle triplet, create rotation matrix 

Given two moment tensors, return the Kagan angle in degrees. 

Scaling relation used by Global CMT catalog for most of its events. 

Convert moment magnitude Mw to scalar moment. 

Get Eulerian angle triplet from rotation matrix. 

Convert scalar moment to moment magnitude Mw. 

Convert northeastdown coordinate vectors to radialtakeoffazimuth. 

Order strikediprake pair post closely to a given reference pair. 

Get the respectively other plane in the doublecouple ambiguity. 

Draw magnitude from Gutenberg Richter distribution. 

Get randomly oriented unit vector. 

Get random rotation matrix. 

Get random strike, dip, rake triplet. 

Build rotation matrix based on axis and angle. 

Create symmetric 3x3 matrix from its 6 nonredundant values. 

Get nonredundant components from symmetric 3x3 matrix. 

Uniquify Eulerian angle triplet. 

Convert anything to moment tensor represented as a NumPy array. 
Classes

Moment tensor object 
 random_axis(rstate=None)[source]¶
Get randomly oriented unit vector.
 Parameters:
rstate –
numpy.random.RandomState
object, can be used to create reproducible pseudorandom sequences
 rotation_from_angle_and_axis(angle, axis)[source]¶
Build rotation matrix based on axis and angle.
 Parameters:
angle – rotation angle [degrees]
axis – orientation of rotation axis, either in spherical coordinates
(theta, phi)
[degrees], or as a unit vector(ux, uy, uz)
.
 random_rotation(x=None)[source]¶
Get random rotation matrix.
A random rotation matrix, drawn from a uniform distribution in the space of rotations is returned, after Avro 1992  “Fast random rotation matrices”.
 Parameters:
x – three (uniform random) numbers in the range [0, 1[ used as input to the distribution transformation. If
None
, random numbers are used. Can be used to create grids of random rotations with uniform density in rotation space.
 random_strike_dip_rake(strikemin=0.0, strikemax=360.0, dipmin=0.0, dipmax=90.0, rakemin=180.0, rakemax=180.0)[source]¶
Get random strike, dip, rake triplet.
Note
Might not produce a homogeneous distribution of mechanisms. Better use
MomentTensor.random_dc()
which is based onrandom_rotation()
.
 to6(m)[source]¶
Get nonredundant components from symmetric 3x3 matrix.
 Returns:
1D NumPy array with entries ordered like
(a_xx, a_yy, a_zz, a_xy, a_xz, a_yz)
 symmat6(a_xx, a_yy, a_zz, a_xy, a_xz, a_yz)[source]¶
Create symmetric 3x3 matrix from its 6 nonredundant values.
 values_to_matrix(values)[source]¶
Convert anything to moment tensor represented as a NumPy array.
Transforms
MomentTensor
objects, tuples, lists and NumPy arrays with 3x3 or 3, 4, 6, or 7 elements into NumPy 3x3 array objects.The
values
argument is interpreted depending on shape and type as follows:(strike, dip, rake)
(strike, dip, rake, magnitude)
(mnn, mee, mdd, mne, mnd, med)
(mnn, mee, mdd, mne, mnd, med, magnitude)
((mnn, mne, mnd), (mne, mee, med), (mnd, med, mdd))
 moment_to_magnitude(moment)[source]¶
Convert scalar moment to moment magnitude Mw.
 Parameters:
moment – scalar moment [Nm]
 Returns:
moment magnitude Mw
Moment magnitude is defined as
where is the scalar moment given in [Nm].
Note
Global CMT uses 10.7333333 instead of 10.7, based on [Kanamori 1977], 10.7 is from [Hanks and Kanamori 1979].
 magnitude_to_moment(magnitude)[source]¶
Convert moment magnitude Mw to scalar moment.
 Parameters:
magnitude – moment magnitude
 Returns:
scalar moment [Nm]
 euler_to_matrix(alpha, beta, gamma)[source]¶
Given euler angle triplet, create rotation matrix
Given coordinate system (x,y,z) and rotated system (xs,ys,zs) the line of nodes is the intersection between the x,y and the xs,ys planes.
 Parameters:
alpha – is the angle between the zaxis and the zsaxis [rad]
beta – is the angle between the xaxis and the line of nodes [rad]
gamma – is the angle between the line of nodes and the xsaxis [rad]
Usage for moment tensors:
m_unrot = numpy.array([[0,0,1],[0,0,0],[1,0,0]]) euler_to_matrix(dip,strike,rake, rotmat) m = num.dot(rotmat.T, num.dot(m_unrot, rotmat))
 unique_euler(alpha, beta, gamma)[source]¶
Uniquify Eulerian angle triplet.
Put Eulerian angle triplet into ranges compatible with
(dip, strike, rake)
conventions in seismology:alpha (dip) : [0, pi/2] beta (strike) : [0, 2*pi) gamma (rake) : [pi, pi)
If
alpha1
is near to zero,beta
is replaced bybeta+gamma
andgamma
is set to zero, to prevent this additional ambiguity.If
alpha
is near topi/2
,beta
is put into the range[0,pi)
.
 as_mt(mt)[source]¶
Convenience function to convert various objects to moment tensor object.
Like
MomentTensor.from_values()
, but does not create a newMomentTensor
object whenmt
already is one.
 ned_to_rta(ned)[source]¶
Convert northeastdown coordinate vectors to radialtakeoffazimuth.
Output coordinate system has coordinates radial, takeoff angle [deg] (downward is zero), and azimuth [deg] (northward is zero).
 Parameters:
ned (
numpy.ndarray
of shape(N, 3)
or(3,)
) – Coordinate vector or array of coordinate vectors (north, east, down). Returns:
Coordinate vector or array of coordinate vectors (radial, takeoff, azimuth)
 Return type:
numpy.ndarray
of shape(N, 3)
or(3,)
 class MomentTensor(m=None, m_up_south_east=None, m_east_north_up=None, strike=0.0, dip=0.0, rake=0.0, scalar_moment=1.0, mnn=None, mee=None, mdd=None, mne=None, mnd=None, med=None, strike1=None, dip1=None, rake1=None, strike2=None, dip2=None, rake2=None, p_axis=None, t_axis=None, magnitude=None, moment=None)[source]¶
Bases:
Object
Moment tensor object
 Parameters:
m – NumPy array in northeastdown convention
m_up_south_east – NumPy array in upsoutheast convention
m_east_north_up – NumPy array in eastnorthup convention
strike,dip,rake – fault plane angles in [degrees]
p_axis,t_axis – initialize doublecouple from p and t axes
scalar_moment – scalar moment in [Nm]
magnitude – moment magnitude Mw
Global CMT catalog moment tensors use the upsoutheast (USE) coordinate system convention with (up), (south), and (east).
 ♦ mnn¶
float
, default:0.0
 ♦ mee¶
float
, default:0.0
 ♦ mdd¶
float
, default:0.0
 ♦ mne¶
float
, default:0.0
 ♦ mnd¶
float
, default:1.0
 ♦ med¶
float
, default:0.0
 ♦ strike1¶
float
, optional
 ♦ dip1¶
float
, optional
 ♦ rake1¶
float
, optional
 ♦ strike2¶
float
, optional
 ♦ dip2¶
float
, optional
 ♦ rake2¶
float
, optional
 ♦ moment¶
float
, optional
 ♦ magnitude¶
float
, optional
 classmethod random_dc(x=None, scalar_moment=1.0, magnitude=None)[source]¶
Create random oriented doublecouple moment tensor
The rotations used are uniformly distributed in the space of rotations.
 classmethod random_mt(x=None, scalar_moment=1.0, magnitude=None)[source]¶
Create random moment tensor
Moment tensors produced by this function appear uniformly distributed when shown in a Hudson’s diagram. The rotations used are uniformly distributed in the space of rotations.
 classmethod from_values(values)[source]¶
Alternative constructor for moment tensor objects
This constructor takes a
MomentTensor
object, a tuple, list or NumPy array with 3x3 or 3, 4, 6, or 7 elements to build a Moment tensor object.The
values
argument is interpreted depending on shape and type as follows:(strike, dip, rake)
(strike, dip, rake, magnitude)
(mnn, mee, mdd, mne, mnd, med)
(mnn, mee, mdd, mne, mnd, med, magnitude)
((mnn, mne, mnd), (mne, mee, med), (mnd, med, mdd))
MomentTensor
object
 p_axis()[source]¶
Get direction of p axis.
Use
ned_to_rta()
to get takeoff angle and azimuth of the returned vectors. Returns:
Direction of P (pressure) axis as a (north, east, down) vector.
 Return type:
numpy.ndarray
of shape(3,)
 t_axis()[source]¶
Get direction of t axis.
Use
ned_to_rta()
to get takeoff angle and azimuth of the returned vectors. Returns:
Direction of T (tension) axis as a (north, east, down) vector.
 Return type:
numpy.ndarray
of shape(3,)
 null_axis()[source]¶
Get direction of the null axis.
Use
ned_to_rta()
to get takeoff angle and azimuth of the returned vectors. Returns:
Direction of null (B) axis as a (north, east, down) vector.
 Return type:
numpy.ndarray
of shape(3,)
 eigenvals()[source]¶
Get the eigenvalues of the moment tensor in ascending order.
 Returns:
(ep, en, et)
 eigensystem()[source]¶
Get the eigenvalues and eigenvectors of the moment tensor.
 Returns:
(ep, en, et, vp, vn, vt)
 m6_up_south_east()[source]¶
Get moment tensor in upsoutheast convention as a sixelement array.
 Returns:
(muu, mss, mee, mus, mue, mse)
 m6_east_north_up()[source]¶
Get moment tensor in eastnorthup convention as a sixelement array.
 Returns:
(mee, mnn, muu, men, meu, mnu)
 scalar_moment()[source]¶
Get the scalar moment of the moment tensor (Frobenius norm based)
The scalar moment is calculated based on the Euclidean (Frobenius) norm (Silver and Jordan, 1982). The scalar moment returned by this function differs from the standard decomposition based definition of the scalar moment for nondoublecouple moment tensors.
 deviatoric()[source]¶
Get deviatoric part of moment tensor.
Returns a new moment tensor object with zero trace.
 standard_decomposition()[source]¶
Decompose moment tensor into isotropic, DC and CLVD components.
Standard decomposition according to e.g. Jost and Herrmann 1989 is returned as:
[ (moment_iso, ratio_iso, m_iso), (moment_dc, ratio_dc, m_dc), (moment_clvd, ratio_clvd, m_clvd), (moment_devi, ratio_devi, m_devi), (moment, 1.0, m) ]
 rotated(rot)[source]¶
Get rotated moment tensor.
 Parameters:
rot – ratation matrix, coordinate system is NED
 Returns:
new
MomentTensor
object
 random_rotated(angle_std=None, angle=None, rstate=None)[source]¶
Get distorted MT by rotation around random axis and angle.
 Parameters:
angle_std – angles are drawn from a normal distribution with zero mean and given standard deviation [degrees]
angle – set angle [degrees], only axis will be random
rstate –
numpy.random.RandomState
object, can be used to create reproducible pseudorandom sequences
 Returns:
new
MomentTensor
object
 other_plane(strike, dip, rake)[source]¶
Get the respectively other plane in the doublecouple ambiguity.
 order_like(sdrs, sdrs_ref)[source]¶
Order strikediprake pair post closely to a given reference pair.
 Parameters:
sdrs – tuple,
((strike1, dip1, rake1), (strike2, dip2, rake2))
sdrs_ref – as above but with reference pair
 kagan_angle(mt1, mt2)[source]¶
Given two moment tensors, return the Kagan angle in degrees.
After Kagan (1991) and Tape & Tape (2012).