pyrocko.modelling.okada¶

Elastostatic solutions and boundary element modelling for rectangular dislocation sources.

Functions

`invert_fault_dislocations_bem`(stress_field)	BEM least squares inversion to get fault dislocations given stress field.
`make_okada_coefficient_matrix`(...[, ...])	Build coefficient matrix for given fault patches.

Classes

`AnalyticalRectangularSource`(**kwargs)	Rectangular analytical source model.
`AnalyticalSource`(**kwargs)	Base class for analytical source models.
`OkadaPatch`([parent])	Okada source with additional 2D indexes for bookkeeping.
`OkadaSource`(**kwargs)	Rectangular Okada source model.

class AnalyticalSource(**kwargs)[source]¶

Bases: Location

Base class for analytical source models.

♦ name¶: str, optional, default: ''

♦ time¶

pyrocko.util.get_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]¶

Bases: AnalyticalSource

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]¶

Bases: AnalyticalRectangularSource

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]¶

Bases: OkadaSource

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 (bool) – If True, only shear forces are taken into account.
rotate_sdn (bool) – If True, rotate to strike, dip, normal.
nthreads (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 (ndarray: (len(source_list) * 3, len(source_list) * 3)) – Coefficient matrix connecting source patch dislocations and the stress field.
source_list (list of OkadaSource) – Source patches to be used for BEM.
epsilon (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)

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