`modelling`¶

`okada`¶

class AnalyticalSource(**kwargs)[source]¶

Base class for analytical source models.

♦ name¶: str, optional, default: ''

♦ time¶

time_float (pyrocko.guts.Timestamp), optional, default: 0.0

Source origin time

♦ vr¶

float, optional, default: 0.0

Rupture velocity [m/s]

class AnalyticalRectangularSource(**kwargs)[source]¶

Rectangular analytical source model.

Coordinates on the source plane are with respect to the origin point given by (lat, lon, east_shift, north_shift, depth).

♦ strike¶

float, default: 0.0

Strike direction in [deg], measured clockwise from north.

♦ dip¶

float, default: 90.0

Dip angle in [deg], measured downward from horizontal.

♦ rake¶

float, default: 0.0

Rake angle in [deg], measured counter-clockwise from right-horizontal in on-plane view.

♦ al1¶

float, default: 0.0

Left edge source plane coordinate [m].

♦ al2¶

float, default: 0.0

Right edge source plane coordinate [m].

♦ aw1¶

float, default: 0.0

Lower edge source plane coordinate [m].

♦ aw2¶

float, default: 0.0

Upper edge source plane coordinate [m].

♦ slip¶

float, optional, default: 0.0

Slip on the rectangular source area [m].

class OkadaSource(**kwargs)[source]¶

Rectangular Okada source model.

♦ opening¶

float, optional, default: 0.0

Opening of the plane in [m].

♦ poisson¶

float, optional, default: 0.25

Poisson ratio $\nu$ . The Poisson ratio $\nu$ . If set to None, calculated from the Lame’ parameters $\lambda$ and $\mu$ using $\nu = \frac{\lambda}{2(\lambda + \mu)}$ (e.g. Mueller 2007).

♦ lamb¶

float, optional

First Lame parameter $\lambda$ [Pa]. If set to None, it is computed from Poisson ratio $\nu$ and shear modulus $\mu$ . Important: We assume a perfect elastic solid with $K=\frac{5}{3}\mu$ . Through $\nu = \frac{\lambda}{2(\lambda + \mu)}$ this leads to $\lambda = \frac{2 \mu \nu}{1-2\nu}$ .

♦ shearmod¶

float, optional, default: 32000000000.0

Shear modulus $\mu$ [Pa]. If set to None, it is computed from poisson ratio. Important: We assume a perfect elastic solid with $K=\frac{5}{3}\mu$ . Through $\mu = \frac{3K(1-2\nu)}{2(1+\nu)}$ this leads to $\mu = \frac{8(1+\nu)}{1-2\nu}$ .

property seismic_moment¶

Scalar Seismic moment $M_0$ .

Code copied from Kite. It disregards the opening (as for now). We assume $M_0 = mu A D$ .

Important

We assume a perfect elastic solid with $K=\frac{5}{3}\mu$ .

Through $\mu = \frac{3K(1-2\nu)}{2(1+\nu)}$ this leads to $\mu = \frac{8(1+\nu)}{1-2\nu}$ .

Returns:: Seismic moment release.
Return type:: float

property moment_magnitude¶

Moment magnitude $M_\mathrm{w}$ from seismic moment.

We assume $M_\mathrm{w} = {\frac{2}{3}}\log_{10}(M_0) - 10.7$ .

Returns:: Moment magnitude.
Return type:: float

source_patch()[source]¶

Get source location and geometry array for okada_ext.okada input.

The values are defined according to Okada (1992).

Returns:: Source data as input for okada_ext.okada. The order is northing [m], easting [m], depth [m], strike [deg], dip [deg], al1 [m], al2 [m], aw1 [m], aw2 [m].
Return type:: ndarray: (9, )

source_disloc()[source]¶

Get source dislocation array for okada_ext.okada input.

The given slip is splitted into a strike and an updip part based on the source rake.

Returns:: Source dislocation data as input for okada_ext.okada. The order is dislocation in strike [m], dislocation updip [m], opening [m].
Return type:: ndarray: (3, )

discretize(nlength, nwidth, *args, **kwargs)[source]¶

Discretize fault into rectilinear grid of fault patches.

Fault orientation, slip and elastic parameters are passed to the sub-faults unchanged.

Parameters:

nlength (int) – Number of patches in strike direction.
nwidth (int) – Number of patches in down-dip direction.

Returns:

Discrete fault patches.

Return type:

list of OkadaPatch

class OkadaPatch(parent=None, *args, **kwargs)[source]¶

Okada source with additional 2D indexes for bookkeeping.

♦ ix¶

int

Relative index of the patch in x

♦ iy¶

int

Relative index of the patch in y

make_okada_coefficient_matrix(source_patches_list, pure_shear=False, rotate_sdn=True, nthreads=1, variant='normal')[source]¶

Build coefficient matrix for given fault patches.

The boundary element method (BEM) for a discretized fault and the determination of the slip distribution $\Delta u$ from stress drop $\Delta \sigma$ is based on $\Delta \sigma = \mathbf{C} \cdot \Delta u$ . Here the coefficient matrix $\mathbf{C}$ is built, based on the displacements from Okada’s solution (Okada, 1992) and their partial derivatives.

Parameters:

source_patches_list (list of OkadaSource.) – Source patches, to be used in BEM.
pure_shear (optional, bool) – If True, only shear forces are taken into account.
rotate_sdn (optional, bool) – If True, rotate to strike, dip, normal.
nthreads (optional, int) – Number of threads.

Returns:

Coefficient matrix for all source combinations.

Return type:

ndarray: (len(source_patches_list) * 3, len(source_patches_list) * 3)

invert_fault_dislocations_bem(stress_field, coef_mat=None, source_list=None, pure_shear=False, epsilon=None, nthreads=1, **kwargs)[source]¶

BEM least squares inversion to get fault dislocations given stress field.

Follows least squares inversion approach by Menke (1989) to calculate dislocations on a fault with several segments from a given stress field. The coefficient matrix connecting stresses and displacements of the fault patches can either be specified by the user (coef_mat) or it is calculated using the solution of Okada (1992) for a rectangular fault in a homogeneous half space (source_list).

Parameters:

stress_field (ndarray: (nsources, 3)) – Stress change [Pa] for each source patch (as stress_field[isource, icomponent] where isource indexes the source patch and icomponent indexes component, ordered (strike, dip, tensile).
coef_mat (optional, ndarray: (len(source_list) * 3, len(source_list) * 3)) – Coefficient matrix connecting source patch dislocations and the stress field.
source_list (optional, list of OkadaSource) – Source patches to be used for BEM.
epsilon (optional, float) – If given, values in coef_mat smaller than epsilon are set to zero.
nthreads (int) – Number of threads allowed.

Returns:

Inverted displacements as displacements[isource, icomponent] where isource indexes the source patch and icomponent indexes component, ordered (strike, dip, tensile).

Return type:

ndarray: (nsources, 3)

`cracksol`¶

Analytical crack solutions for surface displacements and fault dislocations.

class GriffithCrack(**kwargs)[source]¶

Analytical Griffith crack model.

♦ width¶

float, default: 1.0

Width equals to $2 \cdot a$ .

♦ poisson¶

float, default: 0.25

Poisson ratio $\nu$ .

♦ shearmod¶

float, default: 1000000000.0

Shear modulus $\mu$ [Pa].

♦ stressdrop¶

numpy.ndarray (pyrocko.guts_array.Array), default: array([0., 0., 0.])

Stress drop array:[ $\sigma_{3,r} - \sigma_{3,c}$ , $\sigma_{2,r} - \sigma_{2,c}$ , $\sigma_{1,r} - \sigma_{1,c}$ ] $=$ [ $\Delta \sigma_{strike}, \Delta \sigma_{dip}, \Delta \sigma_{tensile}$ ].

property a¶: Half width of the crack in [m].

disloc_infinite2d(x_obs)[source]¶

Calculation of dislocation at crack surface along x2 axis.

Follows equations by Pollard and Segall (1987) to calculate dislocations for an infinite 2D crack extended in x3 direction, opening in x1 direction and the crack extending in x2 direction.

Parameters:: x_obs (ndarray: (N,)) – Observation point coordinates along x2-axis. If $x_{obs} < -a$ or $x_{obs} > a$ , output dislocations are zero.
Returns:: Dislocations at each observation point in strike, dip and tensile direction.
Return type:: ndarray: (N, 3)

disloc_circular(x_obs)[source]¶

Calculation of dislocation at crack surface along x2 axis.

Follows equations by Pollard and Segall (1987) to calculate displacements for a circulat crack extended in x2 and x3 direction and opening in x1 direction.

Parameters:: x_obs (ndarray: (N,)) – Observation point coordinates along axis through crack centre. If $x_{obs} < -a$ or $x_{obs} > a$ , output dislocations are zero.
Returns:: Dislocations at each observation point in strike, dip and tensile direction.
Return type:: ndarray: (N, 3)

displ_infinite2d(x1_obs, x2_obs)[source]¶

Calculation of displacement at crack surface along different axis.

Follows equations by Pollard and Segall (1987) to calculate displacements for an infinite 2D crack extended in x3 direction, opening in x1 direction and the crack tip in x2 direction.

Parameters:

x1_obs (ndarray: (M,)) – Observation point coordinates along x1-axis. If $x1_obs = 0.$ , displacment is calculated along x2-axis.
x2_obs (ndarray: (N,)) – Observation point coordinates along x2-axis. If $x2_obs = 0.$ , displacment is calculated along x1-axis.

Returns:

Displacements at each observation point in strike, dip and tensile direction.

Return type:

ndarray: (M, 3) or (N, 3)

`eikonal`¶

eikonal_solver_fmm_cartesian(speeds, times, delta)[source]¶

Solve eikonal equation in 2D or 3D using the fast marching method.

This function implements the fast marching method (FMM) by [sethian1996].

Parameters:

speeds (2D or 3D numpy.ndarray) – Velocities at the grid nodes.
times (2D or 3D numpy.ndarray, same shape as speeds) – Arrival times (input and output). The solution is obtained at nodes where times is set to a negative value. Values of zero, or positive values are used as seeding points.
delta (float) – Grid spacing.

[sethian1996]

Sethian, James A. “A fast marching level set method for monotonically advancing fronts.” Proceedings of the National Academy of Sciences 93.4 (1996): 1591-1595. https://doi.org/10.1073/pnas.93.4.1591

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modelling¶

okada¶

cracksol¶

eikonal¶

Navigation

`modelling`¶

`okada`¶

`cracksol`¶

`eikonal`¶