esys.downunder Package

Data inversion module built on escript

Classes

class esys.downunder.AbstractMinimizer(J=None, m_tol=0.0001, J_tol=None, imax=300)

Base class for function minimization methods.

__init__(J=None, m_tol=0.0001, J_tol=None, imax=300)

Initializes a new minimizer for a given cost function.

Parameters

  • J (CostFunction) – the cost function to be minimized

  • m_tol (float) – terminate interations when relative change of the level set function is less than or equal m_tol

getCostFunction()

return the cost function to be minimized

Return type

CostFunction

getOptions()

Returns a dictionary of minimizer-specific options.

getResult()

Returns the result of the minimization.

logSummary()

Outputs a summary of the completed minimization process to the logger.

run(x0)

Executes the minimization algorithm for f starting with the initial guess x0.

Returns

the result of the minimization

setCallback(callback)

Sets a callback function to be called after every iteration. It is up to the specific implementation what arguments are passed to the callback. Subclasses should at least pass the current iteration number k, the current estimate x, and possibly f(x), grad f(x), and the current error.

setCostFunction(J)

set the cost function to be minimized

Parameters

J (CostFunction) – the cost function to be minimized

setMaxIterations(imax)

Sets the maximum number of iterations before the minimizer terminates.

setOptions(**opts)

Sets minimizer-specific options. For a list of possible options see getOptions().

setTolerance(m_tol=0.0001, J_tol=None)

Sets the tolerance for the stopping criterion. The minimizer stops when an appropriate norm is less than m_tol.

class esys.downunder.AcousticVelocityMapping(V_prior, Q_prior)

Maps a p-velocity and Q-index to slowness square sigma=(V*(1-i*1/(2*Q))^{-2} in the form sigma=e^{Mr+m[0])}*( cos(Mi+m[1])) + i * sin(Mi+m[1])

__init__(V_prior, Q_prior)

initializes the mapping

Parameters

  • V_prior – a-priori p-wave velocity

  • Q_prior – a-priori Q-index (must be positive)

getDerivative(m)

returns the value for the derivative of the mapping for m

getInverse(s)

returns the value of the inverse of the mapping for s

getValue(m)

returns the value of the mapping for m

class esys.downunder.AcousticWaveForm(domain, omega, w, data, F, coordinates=None, fixAtBottom=False, tol=1e-10, saveMemory=True, scaleF=True)

Forward Model for acoustic waveform inversion in the frequency domain. It defines a cost function:

Math

defect = 1/2 integrate( ( w * ( a * u - data ) ) ** 2 )

where w are weighting factors, data are the measured data (as a 2-comp vector of real and imaginary part) for real frequency omega, and u is the corresponding result produced by the forward model. u (as a 2-comp vector) is the solution of the complex Helmholtz equation for frequency omega, source F and complex, inverse, squared p-velocity sigma:

Math

-u_{ii} - omega**2 * sigma * u = F

It is assumed that the exact scale of source F is unknown and the scaling factor a of F is calculated by minimizing the defect.

__init__(domain, omega, w, data, F, coordinates=None, fixAtBottom=False, tol=1e-10, saveMemory=True, scaleF=True)

initializes a new forward model with acoustic wave form inversion.

Parameters

  • domain (Domain) – domain of the model

  • w (Scalar) – weighting factors

  • data (escript.Data of shape (2,)) – real and imaginary part of data

  • F (escript.Data of shape (2,)) – real and imaginary part of source given at Dirac points, on surface or at volume.

  • coordinates (ReferenceSystem or SpatialCoordinateTransformation) – defines coordinate system to be used (not supported yet)

  • tol (positive float) – tolerance of underlying PDE

  • saveMemory (bool) – if true stiffness matrix is deleted after solution of PDE to minimize memory requests. This will require more compute time as the matrix needs to be reallocated.

  • scaleF (bool) – if true source F is scaled to minimize defect.

  • fixAtBottom (bool) – if true pressure is fixed to zero at the bottom of the domain

getArguments(sigma)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

sigma (escript.Data of shape (2,)) – a suggestion for complex 1/V**2

Returns

solution, uTar, uTai, uTu

Return type

escript.Data of shape (2,), 3 x float

getCoordinateTransformation()

returns the coordinate transformation being used

Return type

CoordinateTransformation

getDefect(sigma, u, uTar, uTai, uTu)

Returns the defect value.

Parameters

  • sigma (escript.Data of shape (2,)) – a suggestion for complex 1/V**2

  • u (escript.Data of shape (2,)) – a u vector

  • uTar (float) – equals integrate( w  * (data[0]*u[0]+data[1]*u[1]))

  • uTai – equals integrate( w  * (data[1]*u[0]-data[0]*u[1]))

  • uTu (float) – equals integrate( w  * (u,u))

Return type

float

getDomain()

Returns the domain of the forward model.

Return type

Domain

getGradient(sigma, u, uTar, uTai, uTu)

Returns the gradient of the defect with respect to density.

Parameters

  • sigma (escript.Data of shape (2,)) – a suggestion for complex 1/V**2

  • u (escript.Data of shape (2,)) – a u vector

  • uTar (float) – equals integrate( w  * (data[0]*u[0]+data[1]*u[1]))

  • uTai – equals integrate( w  * (data[1]*u[0]-data[0]*u[1]))

  • uTu (float) – equals integrate( w  * (u,u))

getSourceScaling(u)

returns the scaling factor s required to rescale source F to minimize defect |s * u- data|^2

Parameters

u (escript.Data of shape (2,)) – value of pressure solution (real and imaginary part)

Return type

complex

getSurvey(index=None)

Returns the pair (data, weight)

If argument index is ignored.

rescaleWeights(scale=1.0, sigma_scale=1.0)

rescales the weights such that

Math

integrate( ( w omega**2 * sigma_scale * data * ((1/L_j)**2)**-1) +1 )/(data*omega**2 * ((1/L_j)**2)**-1) * sigma_scale )=scale

Parameters

  • scale (positive float) – scale of data weighting factors

  • sigma_scale (Scalar) – scale of 1/vp**2 velocity.

setUpPDE()

Creates and returns the underlying PDE.

Return type

lpde.LinearPDE

class esys.downunder.ArithmeticTuple(*args)

Tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.

Example of usage:

from esys.escript import Data
from numpy import array
a=eData(...)
b=array([1.,4.])
x=ArithmeticTuple(a,b)
y=5.*x
__init__(*args)

Initializes object with elements args.

Parameters

args – tuple of objects that support inplace add (x+=y) and scaling (x=a*y)

class esys.downunder.BoundedRangeMapping(s_min=0, s_max=1)

Maps an unbounded parameter to a bounded range. The mapping is smooth and continuous.

__init__(s_min=0, s_max=1)

Initialize self. See help(type(self)) for accurate signature.

getDerivative(m)

returns the value for the derivative of the mapping for m

getInverse(s)

returns the value of the inverse of the mapping for s

getValue(m)

returns the value of the mapping for m

class esys.downunder.CartesianReferenceSystem(name='CARTESIAN')

Identifies the Cartesian coordinate system

__init__(name='CARTESIAN')

set up Cartesian coordinate system

createTransformation(domain)

creates an appropriate coordinate transformation on a given domain

Parameters

domain (esys.escript.AbstractDomain) – domain of transformation

Return type

SpatialCoordinateTransformation

isCartesian()

returns if the reference system is Cartesian

Return type

bool

isTheSame(other)

test if argument other defines the same reference system

Parameters

other (ReferenceSystem) – a second reference system

Returns

True if other is a CartesianReferenceSystem instance.

Return type

bool

Note

every two CartesianReferenceSystem instances are considered as being the same.

class esys.downunder.CostFunction

A function f(x) that can be minimized (base class).

Example of usage:

cf=DerivedCostFunction()
# ... calculate x ...
args=cf.getArguments(x) # this could be potentially expensive!
f=cf.getValue(x, *args)
# ... it could be required to update x without using the gradient...
# ... but then ...
gf=cf.getGradient(x, *args)

The class distinguishes between the representation of the solution x (x-type) and the gradients (r-type).

Note

The provides_inverse_Hessian_approximation class member should be set to True in subclasses that provide a valid implementation of getInverseHessianApproximation()

__init__()

Constructor. Initializes logger.

getArguments(x)

returns precalculated values that are shared in the calculation of f(x) and grad f(x) and the Hessian operator. The default implementation returns an empty tuple.

Note

The tuple returned by this call will be passed back to this CostFunction in other calls(eg: getGradient). Its contents are not specified at this level because no code, other than the CostFunction which created it, will be interacting with it. That is, the implementor can put whatever information they find useful in it.

Parameters

x (x-type) – location of derivative

Return type

tuple

getDualProduct(x, r)

returns the dual product of x and r

Return type

float

getGradient(x, *args)

returns the gradient of f at x using the precalculated values for x.

Parameters

  • x (x-type) – location of derivative

  • args – pre-calculated values for x from getArguments()

Return type

r-type

getInverseHessianApproximation(x, r, *args)

returns an approximative evaluation p of the inverse of the Hessian operator of the cost function for a given gradient r at a given location x: H(x) p = r

Parameters

  • x (x-type) – location of Hessian operator to be evaluated

  • r (r-type) – a given gradient

  • args – pre-calculated values for x from getArguments()

Return type

x-type

Note

In general it is assumed that the Hessian H(x) needs to be calculated in each call for a new location x. However, the solver may suggest that this is not required, typically when the iteration is close to completeness.

Note

Subclasses that implement this method should set the class variable provides_inverse_Hessian_approximation to True to enable the solver to call this method.

getNorm(x)

returns the norm of x

Return type

float

getValue(x, *args)

returns the value f(x) using the precalculated values for x.

Parameters

x (x-type) – a solution approximation

Return type

float

provides_inverse_Hessian_approximation = False
updateHessian()

notifies the class that the Hessian operator needs to be updated. This method is called by the solver class.

class esys.downunder.DataSource(reference_system=None, tags=[])

A class that provides survey data for the inversion process. This is an abstract base class that implements common functionality. Methods to be overwritten by subclasses are marked as such. This class assumes 2D data which is mapped to a slice of a 3D domain. For other setups override the methods as required.

__init__(reference_system=None, tags=[])

Constructor. Sets some defaults and initializes logger.

Parameters

  • tags (list of almost any type (typically str)) – a list of tags associated with the data set.

  • reference_system (None or ReferenceSystem) – the reference coordinate system

ACOUSTIC = 2
GRAVITY = 0
MAGNETIC = 1
MT = 3
getDataExtents()

returns a tuple of tuples ( (x0, y0), (nx, ny), (dx, dy) ), where

  • x0, y0 = coordinates of data origin

  • nx, ny = number of data points in x and y

  • dx, dy = spacing of data points in x and y

This method must be implemented in subclasses.

getDataType()

Returns the type of survey data managed by this source. Subclasses must return GRAVITY or MAGNETIC or ACOUSTIC as appropriate.

getHeightScale()

returns the height scale factor to convert from meters to the appropriate units of the reference system used.

Return type

float

getReferenceSystem()

returns the reference coordinate system

Return type

ReferenceSystem

getSubsamplingFactor()

Returns the subsampling factor that was set via setSubsamplingFactor (see there).

getSurveyData(domain, origin, NE, spacing)

This method is called by the DomainBuilder to retrieve the survey data as Data objects on the given domain.

Subclasses should return one or more Data objects with survey data interpolated on the given escript domain. The exact return type depends on the type of data.

Parameters

  • domain (esys.escript.Domain) – the escript domain to use

  • origin (tuple or list) – the origin coordinates of the domain

  • NE (tuple or list) – the number of domain elements in each dimension

  • spacing (tuple or list) – the cell sizes (node spacing) in the domain

getTags()

returns the list of tags

Return type

list

getUtmZone()

All data source coordinates are converted to UTM (Universal Transverse Mercator) in order to have useful domain extents. Subclasses should implement this method and return the UTM zone number of the projected coordinates.

hasTag(tag)

returns true if the data set has tag tag

Return type

bool

setSubsamplingFactor(f)

Sets the data subsampling factor (default=1).

The factor is applied in all dimensions. For example a 2D dataset with 300 x 150 data points will be reduced to 150 x 75 when a subsampling factor of 2 is used. This becomes important when adding data of varying resolution to a DomainBuilder.

class esys.downunder.DcRes(domain, locator, delphiIn, sampleTags, phiPrimary, sigmaPrimary, w=1.0, coordinates=None, tol=1e-08, saveMemory=True, b=None)

Forward Model for DC resistivity, with a given source pair. The cost function is defined as:

Math

defect = 1/2 (sum_s sum_pq w_pqs * ((phi_sp-phi_sq)-v_pqs)**2

__init__(domain, locator, delphiIn, sampleTags, phiPrimary, sigmaPrimary, w=1.0, coordinates=None, tol=1e-08, saveMemory=True, b=None)

setup new forward model

Parameters

  • domain – the domain of the model

  • locator – contains locator to the measurement pairs

  • sampleTags (list of tuples) – tags of measurement points from which potential differences will be calculated.

  • phiPrimary (Scalar) – primary potential.

Type

escript domain

Type

list of Locator

Param

delphiIn: this is v_pq, the potential difference for the current source and a set of measurement pairs. A list of measured potential differences is expected. Note this should be the secondary potential only.

getArguments(sigma)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

sigma (Data of shape (1,)) – conductivity

Returns

phi

Return type

Data of shape (1,)

getCoordinateTransformation()

returns the coordinate transformation being used

Return type

CoordinateTransformation

getDefect(sigma, phi, loc_phi)

Returns the defect value.

Parameters

  • sigma (Data of shape (1,)) – a suggestion for conductivity

  • phi (Data of shape (1,)) – potential field

Return type

float

getDomain()

Returns the domain of the forward model.

Return type

Domain

getGradient(sigma, phi, loc_phi)

Returns the gradient of the defect with respect to density.

Parameters

  • sigma (Data of shape (1,)) – a suggestison for conductivity

  • phi (Data of shape (1,)) – potential field

getPrimaryPotential()

returns the primary potential :rtype: Data

setUpPDE()

Return the underlying PDE.

Return type

LinearPDE

class esys.downunder.DcResistivityForward

This class allows for the solution of dc resistivity forward problems via the calculation of a primary and secondary potential. Conductivity values are to be provided for the primary problem which is a homogeneous half space of a chosen conductivity and for the secondary problem which typically varies it conductivity spatially across the domain. The primary potential acts as a reference point typically based of some know reference conductivity however any value will suffice. The primary potential is implemented to avoid the use of dirac delta functions.

__init__()

This is a skeleton class for all the other forward modeling classes.

checkBounds()
getApparentResistivity()
getElectrodes()

retuns the list of electrodes with locations

getPotential()
class esys.downunder.DensityMapping(domain, z0=None, rho0=None, drho=None, beta=None)

Density mapping with depth weighting

rho = rho0 + drho * ( (x_2 - z0)/l_z)^(beta/2) ) * m

__init__(domain, z0=None, rho0=None, drho=None, beta=None)

initializes the mapping

Parameters

  • domain (Domain) – domain of the mapping

  • z0 (scalar) – depth weighting offset. If not present no depth scaling is applied.

  • rho0 (scalar) – reference density, defaults to 0

  • drho (scalar) – density scale. By default density of granite = 2750kg/m**3 is used.

  • beta (float) – depth weighting exponent, defaults to 2

class esys.downunder.DipoleDipoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

DipoleDipoleSurvey forward modeling

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

This is a skeleton class for all the other forward modeling classes.

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

class esys.downunder.DomainBuilder(dim=3, reference_system=None)

This class is responsible for constructing an escript Domain object with suitable extents and resolution for survey data (DataSource objects) that are added to it.

The domain covers a region above and below the Earth surface. The East-West direction is used as the x- or longitudinal or x[0] direction, the North-South direction is used as the y- or latitudinal or x[1] direction, the vertical direction is denoted by z or radial or x[2] direction. The corresponding terms are used synonymously.

__init__(dim=3, reference_system=None)

Constructor.

Parameters

  • dim (int) – Dimensionality (2 or 3) of the target domain. This has implications for the survey data than can be added. By default a 3D domain is created.

  • reference_system (ReferenceSystem) – reference coordinate system. By default the Cartesian coordinate system is used.

addSource(source)

Adds a survey data provider to the domain builder. An exception is raised if the domain has already been built. An exception is also reported if the reference system used is cartesian and the UTM zone of source does not match the UTM zone of sources already added to the domain builder (see Inversion Cookbook for more information). The dimensionality of the data source must be compatible with this domain builder. That is, the dimensionality of the data must be one less than the dimensionality of the domain (specified in the constructor).

Parameters

source (DataSource) – The data source to be added. Its reference system needs to match the reference system of the DomainBuilder.

fixDensityBelow(depth=None)

Defines the depth below which the density anomaly is set to a given value. If no value is given zero is assumed.

Parameters

depth (float) – depth below which the density is fixed. If not set, no constraint at depth is applied.

fixSusceptibilityBelow(depth=None)

Defines the depth below which the susceptibility anomaly is set to a given value. If no value is given zero is assumed.

Parameters

depth (float) – depth below which the susceptibility is fixed. If not set, no constraint at depth is applied.

fixVelocityBelow(depth=None)

Defines the depth below which the velocity and Q index is set to a given value. If no value is given zero is assumed.

Parameters

depth (float) – depth below which the velocity is fixed. If not set, no constraint at depth is applied.

getBackgroundMagneticFluxDensity()

Returns the background magnetic flux density.

getDomain()

Returns a domain that spans the data area plus padding.

The domain is created the first time this method is called, subsequent calls return the same domain so anything that affects the domain (such as padding) needs to be set beforehand.

Returns

The escript domain for this data source

Return type

esys.escript.Domain

getGravitySurveys()

Returns a list of gravity surveys, see getSurveys for details.

getMagneticSurveys()

Returns a list of magnetic surveys, see getSurveys for details.

getReferenceSystem()

returns the reference coordinate system

Return type

ReferenceSystem

getSetDensityMask()

Returns the density mask data object which is non-zero for cells whose density value is fixed, zero otherwise.

getSetSusceptibilityMask()

Returns the susceptibility mask data object which is non-zero for cells whose susceptibility value is fixed, zero otherwise.

getSurveys(datatype, tags=None)

Returns a list of Data objects for all surveys of type datatype available to this domain builder. If a list of tags is given only data sources whose tag matches the tag list are returned.

Returns

List of surveys which are tuples (anomaly,error).

Return type

list

getTags()

returns a list of all tags in use by the attached data sources. The list may be empty.

setBackgroundMagneticFluxDensity(B)

Sets the background magnetic flux density B=(B_East, B_North, B_Vertical)

setElementPadding(pad_x=None, pad_y=None, pad_lat=None, pad_lon=None)

Sets the amount of padding around the dataset in number of elements (cells).

When the domain is constructed pad_x (pad_y) elements are added on each side of the x- (y-) dimension. The arguments must be non-negative.

Parameters

  • pad_x (int) – Padding per side in x direction (default: no padding)

  • pad_y (int) – Padding per side in y direction (default: no padding)

Note

pad_y is ignored for 2-dimensional datasets.

setFractionalPadding(pad_x=None, pad_y=None, pad_lat=None, pad_lon=None)

Sets the amount of padding around the dataset as a fraction of the dataset side lengths.

For example, calling setFractionalPadding(0.2, 0.1) with a data source of size 10x20 will result in the padded data set size 14x24 (10*(1+2*0.2), 20*(1+2*0.1))

Parameters

  • pad_x (float) – Padding per side in x direction (default: no padding)

  • pad_y (float) – Padding per side in y direction (default: no padding)

  • pad_lat (float) – Padding per side in latitudinal direction (default: no padding)

  • pad_lon (float) – Padding per side in longitudinal direction (default: no padding)

Note

pad_y is ignored for 2-dimensional domains.

setGeoPadding(pad_lat=None, pad_lon=None)

Sets the amount of padding around the dataset in longitude and latitude.

The final domain size will be the extent in the latitudinal (in longitudinal) direction of the dataset plus twice the value of pad_lat (pad_lon). The arguments must be non-negative.

Parameters

  • pad_lat (float in units of degree) – Padding per side in latitudinal direction (default: 0)

  • pad_lon (float in units of degree) – Padding per side in longitudinal direction (default: 0)

Note

pad_lon is ignored for 2-dimensional domains.

Note

this function can only be used if the reference system is not Cartesian

setPadding(pad_x=None, pad_y=None, pad_lat=None, pad_lon=None)

Sets the amount of padding around the dataset in absolute length units.

The final domain size will be the length in x (in y) of the dataset plus twice the value of pad_x (pad_y). The arguments must be non-negative.

Parameters

  • pad_x (float in units of length (meter)) – Padding per side in x direction (default: no padding)

  • pad_y (float in units of length (meter)) – Padding per side in y direction (default: no padding)

Note

pad_y is ignored for 2-dimensional domains.

Note

this function can only be used if the reference system is Cartesian

setVerticalExtents(depth=40000.0, air_layer=10000.0, num_cells=25)

This method sets the target domain parameters for the vertical dimension.

Parameters

  • depth (float) – Depth of the domain (in meters)

  • air_layer (float) – Depth of the layer above sea level (in meters)

  • num_cells (int) – Number of domain elements for the entire vertical dimension

class esys.downunder.ErMapperData(data_type, headerfile, datafile=None, altitude=0.0, error=None, scale_factor=None, null_value=None, reference_system=None)

Data Source for ER Mapper raster data. Note that this class only accepts a very specific type of ER Mapper data input and will raise an exception if other data is found.

__init__(data_type, headerfile, datafile=None, altitude=0.0, error=None, scale_factor=None, null_value=None, reference_system=None)
Parameters

  • data_type (int) – type of data, must be GRAVITY or MAGNETIC

  • headerfile (str) – ER Mapper header file (usually ends in .ers)

  • datafile (str) – ER Mapper binary data file name. If not supplied the name of the header file without ‘.ers’ is assumed

  • altitude (float) – altitude of measurements above ground in meters

  • error (float) – constant value to use for the data uncertainties. If a value is supplied, it is scaled by the same factor as the measurements. If not provided the error is assumed to be 2 units for all measurements (i.e. 0.2 mGal and 2 nT for gravity and magnetic, respectively)

  • scale_factor (float) – the measurements and error values are scaled by this factor. By default, gravity data is assumed to be given in 1e-6 m/s^2 (0.1 mGal), while magnetic data is assumed to be in 1e-9 T (1 nT).

  • null_value (float) – value that is used in the file to mark undefined areas. This information is usually included in the file.

  • reference_system (ReferenceSystem) – reference coordinate system to be used. For a Cartesian reference (default) the appropriate UTM transformation is applied.

Note

consistence in the reference coordinate system and the reference coordinate system used in the data source is not checked.

getDataExtents()

returns ( (x0, y0), (nx, ny), (dx, dy) )

getDataType()

Returns the type of survey data managed by this source. Subclasses must return GRAVITY or MAGNETIC or ACOUSTIC as appropriate.

getSurveyData(domain, origin, NE, spacing)

This method is called by the DomainBuilder to retrieve the survey data as Data objects on the given domain.

Subclasses should return one or more Data objects with survey data interpolated on the given escript domain. The exact return type depends on the type of data.

Parameters

  • domain (esys.escript.Domain) – the escript domain to use

  • origin (tuple or list) – the origin coordinates of the domain

  • NE (tuple or list) – the number of domain elements in each dimension

  • spacing (tuple or list) – the cell sizes (node spacing) in the domain

getUtmZone()

All data source coordinates are converted to UTM (Universal Transverse Mercator) in order to have useful domain extents. Subclasses should implement this method and return the UTM zone number of the projected coordinates.

class esys.downunder.ForwardModel

An abstract forward model that can be plugged into a cost function. Subclasses need to implement getDefect(), getGradient(), and possibly getArguments() and ‘getCoordinateTransformation’.

__init__()

Initialize self. See help(type(self)) for accurate signature.

getArguments(x)
getCoordinateTransformation()
getDefect(x, *args)
getGradient(x, *args)
class esys.downunder.ForwardModelWithPotential(domain, w, data, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Base class for a forward model using a potential such as magnetic or gravity. It defines a cost function:

defect = 1/2 sum_s integrate( ( weight_i[s] * ( r_i - data_i[s] ) )**2 )

where s runs over the survey, weight_i are weighting factors, data_i are the data, and r_i are the results produced by the forward model. It is assumed that the forward model is produced through postprocessing of the solution of a potential PDE.

__init__(domain, w, data, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

initializes a new forward model with potential.

Parameters

  • domain (Domain) – domain of the model

  • w (Vector or list of Vector) – data weighting factors

  • data (Vector or list of Vector) – data

  • coordinates (ReferenceSystem or SpatialCoordinateTransformation) – defines coordinate system to be used

  • fixPotentialAtBottom (bool) – if true potential is fixed to zero at the bottom of the domain in addition to the top.

  • tol (positive float) – tolerance of underlying PDE

getCoordinateTransformation()

returns the coordinate transformation being used

Return type

CoordinateTransformation

getData()

Returns the data

Return type

list of Data

getDataFunctionSpace()

Returns the FunctionSpace of the data

Return type

FunctionSpace

getDefectGradient(result)
getDomain()

Returns the domain of the forward model.

Return type

Domain

getMisfitWeights()

Returns the weights of the misfit function

Return type

list of Data

getPDE()

Return the underlying PDE.

Return type

LinearPDE

getSurvey(index=None)

Returns the pair (data_index, weight_index), where data_i is the data of survey i, weight_i is the weighting factor for survey i. If index is None, all surveys will be returned in a pair of lists.

class esys.downunder.FunctionJob(fn, *args, **kwargs)

Takes a python function (with only self and keyword params) to be called as the work method

__init__(fn, *args, **kwargs)

It ignores all of its parameters, except that, it requires the following as keyword arguments

Variables

  • domain – Domain to be used as the basis for all Data and PDEs in this Job.

  • jobid – sequence number of this job. The first job has id=1

work()

Need to be overloaded for the job to actually do anthing. A return value of True indicates this job thinks it is done. A return value of False indicates work still to be done

class esys.downunder.GeodeticCoordinateTransformation(domain, reference=<esys.downunder.coordinates.GeodeticReferenceSystem object>)

A geodetic coordinate transformation

__init__(domain, reference=<esys.downunder.coordinates.GeodeticReferenceSystem object>)

set up the orthogonal coordinate transformation.

Parameters

  • domain (esys.escript.AbstractDomain) – domain in the domain of the coordinate transformation

  • reference (ReferenceSystem) – the reference system

class esys.downunder.GeodeticReferenceSystem(a=6378137.0, f=0.0033528106647474805, angular_unit=0.017453292519943295, height_unit=1000.0, name='WGS84')

Identifies a Geodetic coordinate system

__init__(a=6378137.0, f=0.0033528106647474805, angular_unit=0.017453292519943295, height_unit=1000.0, name='WGS84')

initializes a geodetic reference system

Parameters

  • a (positive double) – semi-major axis in meter

  • f (non-negative double, less than one) – flattening

  • name (str) – name of the reference system

  • angular_unit (positive double) – factor to scale the unit of latitude and longitude to radians.

  • height_unit (positive double) – factor to scale the unit of latitude and longitude to radians.

createTransformation(domain)

creates an appropriate coordinate transformation on a given domain

Parameters

domain (esys.escript.AbstractDomain) – domain of transformation

Return type

SpatialCoordinateTransformation

getAngularUnit()

returns the angular unit

getFlattening()

returns the flattening

getHeightUnit()

returns the height unit

getSemiMajorAxis()

returns the length of semi major axis

getSemiMinorAxis()

returns the length of semi minor axis

isCartesian()

returns if the reference system is Cartesian

Return type

bool

isTheSame(other)

test if other defines the same reference system

Parameters

other (ReferenceSystem) – a second reference system

Returns

True if other defines then same reference system

Return type

bool

Note

two GeodeticReferenceSystem are considered to be the same if the use the same semi major axis, the same flattening and the same angular unit.

class esys.downunder.GravityInversion(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

Driver class to perform an inversion of Gravity (Bouguer) anomaly data. The class uses the standard Regularization class for a single level set function, DensityMapping mapping, and the gravity forward model GravityModel.

__init__(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

creates an instance of an inversion driver.

Parameters

  • solverclass ('AbstractMinimizer'.) – class of the solver to be used.

  • self_demagnetization – if True self-demagnitization is applied.

  • magnetic_intensity_data – if True magnetic intensity is used in the cost function.

setInitialGuess(rho=None)

set the initial guess rho for density the inversion iteration. If no rho present then an appropriate initial guess is chosen.

Parameters

rho (Scalar) – initial value for the density anomaly.

setup(domainbuilder, rho0=None, drho=None, z0=None, beta=None, w0=None, w1=None, rho_at_depth=None)

Sets up the inversion parameters from a DomainBuilder.

Parameters

  • domainbuilder (DomainBuilder) – Domain builder object with gravity source(s)

  • rho0 (float or Scalar) – reference density, see DensityMapping. If not specified, zero is used.

  • drho (float or Scalar) – density scale, see DensityMapping. If not specified, 2750kg/m^3 is used.

  • z0 (float or Scalar) – reference depth for depth weighting, see DensityMapping. If not specified, zero is used.

  • beta (float or Scalar) – exponent for depth weighting, see DensityMapping. If not specified, zero is used.

  • w0 (Scalar or float) – weighting factor for level set term regularization. If not set zero is assumed.

  • w1 (Vector or list of float) – weighting factor for the gradient term in the regularization. If not set zero is assumed

  • rho_at_depth (float or None) – value for density at depth, see DomainBuilder.

siloWriterCallback(k, x, Jx, g_Jx, norm_dJ=None, norm_dx=None)

callback function that can be used to track the solution

Parameters

  • k – iteration count

  • x – current approximation

  • Jx – value of cost function

  • g_Jx – gradient of f at x

class esys.downunder.GravityModel(domain, w, g, gravity_constant=6.6742e-11, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Forward Model for gravity inversion as described in the inversion cookbook.

__init__(domain, w, g, gravity_constant=6.6742e-11, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Creates a new gravity model on the given domain with one or more surveys (w, g).

Parameters

  • domain (Domain) – domain of the model

  • w (Vector or list of Vector) – data weighting factors

  • g (Vector or list of Vector) – gravity anomaly data

  • coordinates (ReferenceSystem` or SpatialCoordinateTransformation) – defines coordinate system to be used

  • tol (positive float) – tolerance of underlying PDE

  • fixPotentialAtBottom (bool) – if true potential is fixed to zero at the base of the domain in addition to the top

Note

It is advisable to call rescaleWeights() to rescale weights before starting the inversion.

getArguments(rho)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

rho (Scalar) – a suggestion for the density distribution

Returns

gravity potential and corresponding gravity field.

Return type

Scalar, Vector

getDefect(rho, phi, gravity_force)

Returns the value of the defect

Parameters

  • rho (Scalar) – density distribution

  • phi (Scalar) – corresponding potential

  • gravity_force (Vector) – gravity force

Return type

float

getGradient(rho, phi, gravity_force)

Returns the gradient of the defect with respect to density.

Parameters

  • rho (Scalar) – density distribution

  • phi (Scalar) – corresponding potential

  • gravity_force (Vector) – gravity force

Return type

Scalar

getPotential(rho)

Calculates the gravity potential for a given density distribution.

Parameters

rho (Scalar) – a suggestion for the density distribution

Returns

gravity potential

Return type

Scalar

rescaleWeights(scale=1.0, rho_scale=1.0)

rescales the weights such that

sum_s integrate( ( w_i[s] *g_i[s]) (w_j[s]*1/L_j) * L**2 * 4*pi*G*rho_scale )=scale

Parameters

  • scale (positive float) – scale of data weighting factors

  • rho_scale (Scalar) – scale of density.

class esys.downunder.HTIWave(domain, v_p, v_s, wavelet, source_tag, source_vector=[1.0, 0.0, 0.0], eps=0.0, gamma=0.0, delta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=None, absorption_cut=0.01, lumping=True, disable_fast_assemblers=False)

Solving the HTI wave equation (along the x_0 axis)

Note

In case of a two dimensional domain a horizontal domain is considered, i.e. the depth component is dropped.

__init__(domain, v_p, v_s, wavelet, source_tag, source_vector=[1.0, 0.0, 0.0], eps=0.0, gamma=0.0, delta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=None, absorption_cut=0.01, lumping=True, disable_fast_assemblers=False)

initialize the VTI wave solver

Parameters

  • domain (Domain) – domain of the problem

  • v_p (escript.Scalar) – vertical p-velocity field

  • v_s (escript.Scalar) – vertical s-velocity field

  • wavelet (Wavelet) – wavelet to describe the time evolution of source term

  • source_tag ('str' or 'int') – tag of the source location

  • source_vector – source orientation vector

  • eps – first Thompsen parameter

  • delta – second Thompsen parameter

  • gamma – third Thompsen parameter

  • rho – density

  • dt – time step size. If not present a suitable time step size is calculated.

  • u0 – initial solution. If not present zero is used.

  • v0 – initial solution change rate. If not present zero is used.

  • absorption_zone – thickness of absorption zone

  • absorption_cut – boundary value of absorption decay factor

  • lumping – if True mass matrix lumping is being used. This is accelerates the computing but introduces some diffusion.

  • disable_fast_assemblers – if True, forces use of slower and more general PDE assemblers

setQ(q)

sets the PDE q value

Parameters

q – the value to set

class esys.downunder.InversionCostFunction(regularization, mappings, forward_models)

Class to define cost function J(m) for inversion with one or more forward models based on a multi-valued level set function m:

J(m) = J_reg(m) + sum_f mu_f * J_f(p)

where J_reg(m) is the regularization and cross gradient component of the cost function applied to a level set function m, J_f(p) are the data defect cost functions involving a physical forward model using the physical parameter(s) p and mu_f is the trade-off factor for model f.

A forward model depends on a set of physical parameters p which are constructed from components of the level set function m via mappings.

Example 1 (single forward model):

m=Mapping() f=ForwardModel() J=InversionCostFunction(Regularization(), m, f)

Example 2 (two forward models on a single valued level set)

m0=Mapping() m1=Mapping() f0=ForwardModel() f1=ForwardModel()

J=InversionCostFunction(Regularization(), mappings=[m0, m1], forward_models=[(f0, 0), (f1,1)])

Example 3 (two forward models on 2-valued level set)

m0=Mapping() m1=Mapping() f0=ForwardModel() f1=ForwardModel()

J=InversionCostFunction(Regularization(self.numLevelSets=2), mappings=[(m0,0), (m1,0)], forward_models=[(f0, 0), (f1,1)])

Note

If provides_inverse_Hessian_approximation is true, then the class provides an approximative inverse of the Hessian operator.

__init__(regularization, mappings, forward_models)

constructor for the cost function. Stores the supplied object references and sets default weights.

Parameters

  • regularization (Regularization) – the regularization part of the cost function

  • mappings (Mapping or list) – the mappings to calculate physical parameters from the regularization. This is a list of 2-tuples (map, i) where the first component map defines a Mapping and the second component i defines the index of the component of level set function to be used to calculate the mapping. Items in the list may also be just Mapping objects in which case the entire level set function is fed into the Mapping (typically used for a single-component level set function.

  • forward_models – the forward models involved in the calculation of the cost function. This is a list of 2-tuples (f, ii) where the first component f defines a ForwardModel and the second component ii a list of indexes referring to the physical parameters in the mappings list. The 2-tuple can be replaced by a ForwardModel if the mappings list has a single entry.

  • forward_models – ForwardModel or list

createLevelSetFunction(*props)

returns an instance of an object used to represent a level set function initialized with zeros. Components can be overwritten by physical properties props. If present entries must correspond to the mappings arguments in the constructor. Use None for properties for which no value is given.

getComponentValues(m, *args)
getDomain()

returns the domain of the cost function

Return type

Domain

getForwardModel(idx=None)

returns the idx-th forward model.

Parameters

idx (int) – model index. If cost function contains one model only idx can be omitted.

getNumTradeOffFactors()

returns the number of trade-off factors being used including the trade-off factors used in the regularization component.

Return type

int

getProperties(m, return_list=False)

returns a list of the physical properties from a given level set function m using the mappings of the cost function.

Parameters

  • m (Data) – level set function

  • return_list (bool) – if True a list is returned.

Return type

list of Data

getRegularization()

returns the regularization

Return type

Regularization

getTradeOffFactors(mu=None)

returns a list of the trade-off factors.

Return type

list of float

getTradeOffFactorsModels()

returns the trade-off factors for the forward models

Return type

float or list of float

provides_inverse_Hessian_approximation = True
setTradeOffFactors(mu=None)

sets the trade-off factors for the forward model and regularization terms.

Parameters

mu (list of float) – list of trade-off factors.

setTradeOffFactorsModels(mu=None)

sets the trade-off factors for the forward model components.

Parameters

mu (float in case of a single model or a list of float with the length of the number of models.) – list of the trade-off factors. If not present ones are used.

setTradeOffFactorsRegularization(mu=None, mu_c=None)

sets the trade-off factors for the regularization component of the cost function, see Regularization for details.

Parameters

  • mu – trade-off factors for the level-set variation part

  • mu_c – trade-off factors for the cross gradient variation part

updateHessian()

notifies the class that the Hessian operator needs to be updated.

class esys.downunder.InversionDriver(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

Base class for running an inversion

__init__(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

creates an instance of an inversion driver.

Parameters

  • solverclass ('AbstractMinimizer'.) – class of the solver to be used.

  • self_demagnetization – if True self-demagnitization is applied.

  • magnetic_intensity_data – if True magnetic intensity is used in the cost function.

fixGravityPotentialAtBottom(status=False)

indicates to fix the gravity potential at the bottom to zero (in addition to the top)

Parameters

status (bool) – if True gravity potential at the bottom is set to zero

fixMagneticPotentialAtBottom(status=True)

indicates to fix the magnetic potential at the bottom to zero (in addition to the top)

Parameters

status (bool) – if True magnetic potential at the bottom is set to zero

getCostFunction()

returns the cost function of the inversion.

Return type

‘InversionCostFunction’

getDomain()

returns the domain of the inversion

Return type

Domain

getLevelSetFunction()

returns the level set function as solution of the optimization problem

Return type

Data

getSolver()

The solver to be used in the inversion process. See the minimizers module for available solvers. By default, the L-BFGS minimizer is used.

Return type

‘AbstractMinimizer’.

isSetUp()

returns True if the inversion is set up and is ready to run.

Return type

bool

run()

Executes the inversion.

Returns

physical parameters as result of the inversion

Return type

list of physical parameters or a physical parameter

setCostFunction(costfunction)

sets the cost function of the inversion. This function needs to be called before the inversion iteration can be started.

Parameters

costfunction ('InversionCostFunction') – domain of the inversion

setInitialGuess(*props)

Sets the initial guess for the inversion iteration. By default zero is used.

setSolverCallback(callback=None)

Sets the callback function which is called after every solver iteration.

setSolverMaxIterations(maxiter=None)

Sets the maximum number of solver iterations to run. If maxiter is reached the iteration is terminated and MinimizerMaxIterReached is thrown.

Parameters

maxiter (positive int) – maximum number of iteration steps.

setSolverTolerance(m_tol=None, J_tol=None)

Sets the error tolerance for the solver. An acceptable solution is considered to be found once the tolerance is reached.

Parameters

  • m_tol (float or None) – tolerance for changes to level set function. If None changes to the level set function are not checked for convergence during iteration.

  • J_tol (float or None) – tolerance for changes to cost function. If None changes to the cost function are not checked for convergence during iteration.

Note

if both arguments are None the default setting m_tol=1e-4, J_tol=None is used.

setup(*args, **k_args)

sets up the inversion. The default implementation does nothing but it is advised to call this method before calling run().

class esys.downunder.IsostaticPressure(domain, p0=0.0, level0=0, gravity0=- 9.81, background_density=2670.0, gravity_constant=6.6742e-11, coordinates=None, tol=1e-08)

class to calculate isostatic pressure field correction due to gravity forces

__init__(domain, p0=0.0, level0=0, gravity0=- 9.81, background_density=2670.0, gravity_constant=6.6742e-11, coordinates=None, tol=1e-08)
Parameters

  • domain (Domain) – domain of the model

  • p0 (scalar Data or float) – pressure at level0

  • background_density (float) – defines background_density in kg/m^3

  • coordinates (ReferenceSystem` or SpatialCoordinateTransformation) – defines coordinate system to be used

  • tol (positive float) – tolerance of underlying PDE

  • level0 (float) – pressure for z>=`level0` is set to zero.

  • gravity0 (float) – vertical background gravity at level0

getPressure(g=None, rho=None)

return the pressure for gravity force anomaly g and density anomaly rho

Parameters

  • g (Vector) – gravity anomaly data

  • rho (Scalar) – gravity anomaly data

Returns

pressure distribution

Return type

Scalar

class esys.downunder.Job(*args, **kwargs)

Describes a sequence of work to be carried out in a subworld. The instances of this class used in the subworlds will be constructed by the system. To do specific work, this class should be subclassed and the work() (and possibly __init__ methods overloaded). The majority of the work done by the job will be in the overloaded work() method. The work() method should retrieve values from the outside using importValue() and pass values to the rest of the system using exportValue(). The rest of the methods should be considered off limits.

__init__(*args, **kwargs)

It ignores all of its parameters, except that, it requires the following as keyword arguments

Variables

  • domain – Domain to be used as the basis for all Data and PDEs in this Job.

  • jobid – sequence number of this job. The first job has id=1

clearExports()

Remove exported values from the map

clearImports()

Remove imported values from their map

declareImport(name)

Adds name to the list of imports

exportValue(name, v)

Make value v available to other Jobs under the label name. name must have already been registered with the SplitWorld instance. For use inside the work() method.

Variables

  • name – registered label for exported value

  • v – value to be imported

importValue(name)

For use inside the work() method.

Variables

name – label for imported value.

setImportValue(name, v)

Use to make a value available to the job (ie called from outside the job)

Variables

  • name – label used to identify this import

  • v – value to be imported

work()

Need to be overloaded for the job to actually do anthing. A return value of True indicates this job thinks it is done. A return value of False indicates work still to be done

class esys.downunder.JointGravityMagneticInversion(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

Driver class to perform a joint inversion of Gravity (Bouguer) and magnetic anomaly data. The class uses the standard Regularization class for two level set functions with cross-gradient correlation, DensityMapping and SusceptibilityMapping mappings, the gravity forward model GravityModel and the linear magnetic forward model MagneticModel.

__init__(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

creates an instance of an inversion driver.

Parameters

  • solverclass ('AbstractMinimizer'.) – class of the solver to be used.

  • self_demagnetization – if True self-demagnitization is applied.

  • magnetic_intensity_data – if True magnetic intensity is used in the cost function.

DENSITY = 0
SUSCEPTIBILITY = 1
setInitialGuess(rho=None, k=None)

set the initial guess rho for density and k for susceptibility for the inversion iteration.

Parameters

  • rho (Scalar) – initial value for the density anomaly.

  • k (Scalar) – initial value for the susceptibility anomaly.

setup(domainbuilder, rho0=None, drho=None, rho_z0=None, rho_beta=None, k0=None, dk=None, k_z0=None, k_beta=None, w0=None, w1=None, w_gc=None, rho_at_depth=None, k_at_depth=None)

Sets up the inversion from an instance domainbuilder of a DomainBuilder. Gravity and magnetic data attached to the domainbuilder are considered in the inversion. If magnetic data are given as scalar it is assumed that values are collected in direction of the background magnetic field.

Parameters

  • domainbuilder (DomainBuilder) – Domain builder object with gravity source(s)

  • rho0 (float or Scalar) – reference density, see DensityMapping. If not specified, zero is used.

  • drho (float or Scalar) – density scale, see DensityMapping. If not specified, 2750kg/m^3 is used.

  • rho_z0 (float or Scalar) – reference depth for depth weighting for density, see DensityMapping. If not specified, zero is used.

  • rho_beta (float or Scalar) – exponent for depth weighting for density, see DensityMapping. If not specified, zero is used.

  • k0 (float or Scalar) – reference susceptibility, see SusceptibilityMapping. If not specified, zero is used.

  • dk (float or Scalar) – susceptibility scale, see SusceptibilityMapping. If not specified, 2750kg/m^3 is used.

  • k_z0 (float or Scalar) – reference depth for depth weighting for susceptibility, see SusceptibilityMapping. If not specified, zero is used.

  • k_beta (float or Scalar) – exponent for depth weighting for susceptibility, see SusceptibilityMapping. If not specified, zero is used.

  • w0 (es.Data or ndarray of shape (2,)) – weighting factors for level set term regularization, see Regularization. If not set zero is assumed.

  • w1 (es.Data or ndarray of shape (2,DIM)) – weighting factor for the gradient term in the regularization see Regularization. If not set zero is assumed

  • w_gc (Scalar or float) – weighting factor for the cross gradient term in the regularization, see Regularization. If not set one is assumed

  • k_at_depth (float or None) – value for susceptibility at depth, see DomainBuilder.

  • rho_at_depth (float or None) – value for density at depth, see DomainBuilder.

siloWriterCallback(k, x, Jx, g_Jx, norm_dJ=None, norm_dx=None)

callback function that can be used to track the solution

Parameters

  • k – iteration count

  • x – current approximation

  • Jx – value of cost function

  • g_Jx – gradient of f at x

class esys.downunder.LinearMapping(a=1.0, p0=0.0)

Maps a parameter by a linear transformation p = a * m + p0

__init__(a=1.0, p0=0.0)

Initialize self. See help(type(self)) for accurate signature.

getDerivative(m)

returns the value for the derivative of the mapping for m

getInverse(p)

returns the value of the inverse of the mapping for s

getTypicalDerivative()

returns a typical value for the derivative

getValue(m)

returns the value of the mapping for m

class esys.downunder.LinearPDE(domain, numEquations=None, numSolutions=None, isComplex=False, debug=False)

This class is used to define a general linear, steady, second order PDE for an unknown function u on a given domain defined through a Domain object.

For a single PDE having a solution with a single component the linear PDE is defined in the following form:

-(grad(A[j,l]+A_reduced[j,l])*grad(u)[l]+(B[j]+B_reduced[j])u)[j]+(C[l]+C_reduced[l])*grad(u)[l]+(D+D_reduced)=-grad(X+X_reduced)[j,j]+(Y+Y_reduced)

where grad(F) denotes the spatial derivative of F. Einstein’s summation convention, ie. summation over indexes appearing twice in a term of a sum performed, is used. The coefficients A, B, C, D, X and Y have to be specified through Data objects in Function and the coefficients A_reduced, B_reduced, C_reduced, D_reduced, X_reduced and Y_reduced have to be specified through Data objects in ReducedFunction. It is also allowed to use objects that can be converted into such Data objects. A and A_reduced are rank two, B, C, X, B_reduced, C_reduced and X_reduced are rank one and D, D_reduced, Y and Y_reduced are scalar.

The following natural boundary conditions are considered:

n[j]*((A[i,j]+A_reduced[i,j])*grad(u)[l]+(B+B_reduced)[j]*u)+(d+d_reduced)*u=n[j]*(X[j]+X_reduced[j])+y

where n is the outer normal field. Notice that the coefficients A, A_reduced, B, B_reduced, X and X_reduced are defined in the PDE. The coefficients d and y are each a scalar in FunctionOnBoundary and the coefficients d_reduced and y_reduced are each a scalar in ReducedFunctionOnBoundary.

Constraints for the solution prescribe the value of the solution at certain locations in the domain. They have the form

u=r where q>0

r and q are each scalar where q is the characteristic function defining where the constraint is applied. The constraints override any other condition set by the PDE or the boundary condition.

The PDE is symmetrical if

A[i,j]=A[j,i] and B[j]=C[j] and A_reduced[i,j]=A_reduced[j,i] and B_reduced[j]=C_reduced[j]

For a system of PDEs and a solution with several components the PDE has the form

-grad((A[i,j,k,l]+A_reduced[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k])[j]+(C[i,k,l]+C_reduced[i,k,l])*grad(u[k])[l]+(D[i,k]+D_reduced[i,k]*u[k] =-grad(X[i,j]+X_reduced[i,j])[j]+Y[i]+Y_reduced[i]

A and A_reduced are of rank four, B, B_reduced, C and C_reduced are each of rank three, D, D_reduced, X_reduced and X are each of rank two and Y and Y_reduced are of rank one. The natural boundary conditions take the form:

n[j]*((A[i,j,k,l]+A_reduced[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k])+(d[i,k]+d_reduced[i,k])*u[k]=n[j]*(X[i,j]+X_reduced[i,j])+y[i]+y_reduced[i]

The coefficient d is of rank two and y is of rank one both in FunctionOnBoundary. The coefficients d_reduced is of rank two and y_reduced is of rank one both in ReducedFunctionOnBoundary.

Constraints take the form

u[i]=r[i] where q[i]>0

r and q are each rank one. Notice that at some locations not necessarily all components must have a constraint.

The system of PDEs is symmetrical if

  • A[i,j,k,l]=A[k,l,i,j]

  • A_reduced[i,j,k,l]=A_reduced[k,l,i,j]

  • B[i,j,k]=C[k,i,j]

  • B_reduced[i,j,k]=C_reduced[k,i,j]

  • D[i,k]=D[i,k]

  • D_reduced[i,k]=D_reduced[i,k]

  • d[i,k]=d[k,i]

  • d_reduced[i,k]=d_reduced[k,i]

LinearPDE also supports solution discontinuities over a contact region in the domain. To specify the conditions across the discontinuity we are using the generalised flux J which, in the case of a system of PDEs and several components of the solution, is defined as

J[i,j]=(A[i,j,k,l]+A_reduced[[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k]-X[i,j]-X_reduced[i,j]

For the case of single solution component and single PDE J is defined as

J[j]=(A[i,j]+A_reduced[i,j])*grad(u)[j]+(B[i]+B_reduced[i])*u-X[i]-X_reduced[i]

In the context of discontinuities n denotes the normal on the discontinuity pointing from side 0 towards side 1 calculated from FunctionSpace.getNormal of FunctionOnContactZero. For a system of PDEs the contact condition takes the form

n[j]*J0[i,j]=n[j]*J1[i,j]=(y_contact[i]+y_contact_reduced[i])- (d_contact[i,k]+d_contact_reduced[i,k])*jump(u)[k]

where J0 and J1 are the fluxes on side 0 and side 1 of the discontinuity, respectively. jump(u), which is the difference of the solution at side 1 and at side 0, denotes the jump of u across discontinuity along the normal calculated by jump. The coefficient d_contact is of rank two and y_contact is of rank one both in FunctionOnContactZero or FunctionOnContactOne. The coefficient d_contact_reduced is of rank two and y_contact_reduced is of rank one both in ReducedFunctionOnContactZero or ReducedFunctionOnContactOne. In case of a single PDE and a single component solution the contact condition takes the form

n[j]*J0_{j}=n[j]*J1_{j}=(y_contact+y_contact_reduced)-(d_contact+y_contact_reduced)*jump(u)

In this case the coefficient d_contact and y_contact are each scalar both in FunctionOnContactZero or FunctionOnContactOne and the coefficient d_contact_reduced and y_contact_reduced are each scalar both in ReducedFunctionOnContactZero or ReducedFunctionOnContactOne.

Typical usage:

p = LinearPDE(dom)
p.setValue(A=kronecker(dom), D=1, Y=0.5)
u = p.getSolution()
__init__(domain, numEquations=None, numSolutions=None, isComplex=False, debug=False)

Initializes a new linear PDE.

Parameters

  • domain (Domain) – domain of the PDE

  • numEquations – number of equations. If None the number of equations is extracted from the PDE coefficients.

  • numSolutions – number of solution components. If None the number of solution components is extracted from the PDE coefficients.

  • debug – if True debug information is printed

checkSymmetry(verbose=True)

Tests the PDE for symmetry.

Parameters

verbose (bool) – if set to True or not present a report on coefficients which break the symmetry is printed.

Returns

True if the PDE is symmetric

Return type

bool

Note

This is a very expensive operation. It should be used for degugging only! The symmetry flag is not altered.

createOperator()

Returns an instance of a new operator.

getFlux(u=None)

Returns the flux J for a given u.

J[i,j]=(A[i,j,k,l]+A_reduced[A[i,j,k,l]]*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])u[k]-X[i,j]-X_reduced[i,j]

or

J[j]=(A[i,j]+A_reduced[i,j])*grad(u)[l]+(B[j]+B_reduced[j])u-X[j]-X_reduced[j]

Parameters

u (Data or None) – argument in the flux. If u is not present or equals None the current solution is used.

Returns

flux

Return type

Data

getRequiredOperatorType()

Returns the system type which needs to be used by the current set up.

getResidual(u=None)

Returns the residual of u or the current solution if u is not present.

Parameters

u (Data or None) – argument in the residual calculation. It must be representable in self.getFunctionSpaceForSolution(). If u is not present or equals None the current solution is used.

Returns

residual of u

Return type

Data

getSolution()

Returns the solution of the PDE.

Returns

the solution

Return type

Data

getSystem()

Returns the operator and right hand side of the PDE.

Returns

the discrete version of the PDE

Return type

tuple of Operator and Data

insertConstraint(rhs_only=False)

Applies the constraints defined by q and r to the PDE.

Parameters

rhs_only (bool) – if True only the right hand side is altered by the constraint

setValue(**coefficients)

Sets new values to coefficients.

Parameters

  • coefficients – new values assigned to coefficients

  • A (any type that can be cast to a Data object on Function) – value for coefficient A

  • A_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient A_reduced

  • B (any type that can be cast to a Data object on Function) – value for coefficient B

  • B_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient B_reduced

  • C (any type that can be cast to a Data object on Function) – value for coefficient C

  • C_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient C_reduced

  • D (any type that can be cast to a Data object on Function) – value for coefficient D

  • D_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient D_reduced

  • X (any type that can be cast to a Data object on Function) – value for coefficient X

  • X_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient X_reduced

  • Y (any type that can be cast to a Data object on Function) – value for coefficient Y

  • Y_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient Y_reduced

  • d (any type that can be cast to a Data object on FunctionOnBoundary) – value for coefficient d

  • d_reduced (any type that can be cast to a Data object on ReducedFunctionOnBoundary) – value for coefficient d_reduced

  • y (any type that can be cast to a Data object on FunctionOnBoundary) – value for coefficient y

  • d_contact (any type that can be cast to a Data object on FunctionOnContactOne or FunctionOnContactZero) – value for coefficient d_contact

  • d_contact_reduced (any type that can be cast to a Data object on ReducedFunctionOnContactOne or ReducedFunctionOnContactZero) – value for coefficient d_contact_reduced

  • y_contact (any type that can be cast to a Data object on FunctionOnContactOne or FunctionOnContactZero) – value for coefficient y_contact

  • y_contact_reduced (any type that can be cast to a Data object on ReducedFunctionOnContactOne or ReducedFunctionOnContactZero) – value for coefficient y_contact_reduced

  • d_dirac (any type that can be cast to a Data object on DiracDeltaFunctions) – value for coefficient d_dirac

  • y_dirac (any type that can be cast to a Data object on DiracDeltaFunctions) – value for coefficient y_dirac

  • r (any type that can be cast to a Data object on Solution or ReducedSolution depending on whether reduced order is used for the solution) – values prescribed to the solution at the locations of constraints

  • q (any type that can be cast to a Data object on Solution or ReducedSolution depending on whether reduced order is used for the representation of the equation) – mask for location of constraints

Raises

IllegalCoefficient – if an unknown coefficient keyword is used

class esys.downunder.Locator(where, x=array([0., 0., 0.]))

Locator provides access to the values of data objects at a given spatial coordinate x.

In fact, a Locator object finds the sample in the set of samples of a given function space or domain which is closest to the given point x.

__init__(where, x=array([0., 0., 0.]))

Initializes a Locator to access values in Data objects on the Doamin or FunctionSpace for the sample point which is closest to the given point x.

Parameters

  • where (escript.FunctionSpace) – function space

  • x (numpy.ndarray or list of numpy.ndarray) – location(s) of the Locator

getFunctionSpace()

Returns the function space of the Locator.

getId(item=None)

Returns the identifier of the location.

getValue(data)

Returns the value of data at the Locator if data is a Data object otherwise the object is returned.

getX()

Returns the exact coordinates of the Locator.

setValue(data, v)

Sets the value of the data at the Locator.

class esys.downunder.MT2DModelTEMode(domain, omega, x, Z_XY, eta=None, w0=1.0, mu=1.2566370614359173e-06, sigma0=0.01, airLayerLevel=None, coordinates=None, Ex_top=1, fixAtTop=False, tol=1e-08, saveMemory=False, directSolver=True)

Forward Model for two dimensional MT model in the TE mode for a given frequency omega. It defines a cost function:

  • defect = 1/2 integrate( sum_s w^s * ( E_x/H_y - Z_XY^s ) ) ** 2 *

where E_x is the horizontal electric field perpendicular to the YZ-domain, horizontal magnetic field H_y=1/(i*omega*mu) * E_{x,z} with complex unit i and permeability mu. The weighting factor w^s is set to

  • w^s(X) = w_0^s *

if length(X-X^s) <= eta and zero otherwise. X^s is the location of impedance measurement Z_XY^s, w_0^s is the level of confidence (eg. 1/measurement error) and eta is level of spatial confidence.

E_x is given as solution of the PDE

  • -E_{x,ii} - i omega * mu * sigma * E_x = 0

where E_x at top and bottom is set to solution for background field. Homogeneous Neuman conditions are assumed elsewhere.

__init__(domain, omega, x, Z_XY, eta=None, w0=1.0, mu=1.2566370614359173e-06, sigma0=0.01, airLayerLevel=None, coordinates=None, Ex_top=1, fixAtTop=False, tol=1e-08, saveMemory=False, directSolver=True)

initializes a new forward model. See base class for a description of the arguments.

getArguments(sigma)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

sigma (Data of shape (2,)) – conductivity

Returns

E_x, E_z

Return type

Data of shape (2,)

getDefect(sigma, Ex, dExdz)

Returns the defect value.

Parameters

  • sigma (Data of shape ()) – a suggestion for conductivity

  • Ex (Data of shape (2,)) – electric field

  • dExdz (Data of shape (2,)) – vertical derivative of electric field

Return type

float

getGradient(sigma, Ex, dExdz)

Returns the gradient of the defect with respect to density.

Parameters

  • sigma (Data of shape ()) – a suggestion for conductivity

  • Ex (Data of shape (2,)) – electric field

  • dExdz (Data of shape (2,)) – vertical derivative of electric field

class esys.downunder.MT2DModelTMMode(domain, omega, x, Z_YX, eta=None, w0=1.0, mu=1.2566370614359173e-06, sigma0=0.01, airLayerLevel=None, coordinates=None, tol=1e-08, saveMemory=False, directSolver=True)

Forward Model for two-dimensional MT model in the TM mode for a given frequency omega. It defines a cost function:

  • defect = 1/2 integrate( sum_s w^s * ( rho*H_x/Hy - Z_YX^s ) ) ** 2 *

where H_x is the horizontal magnetic field perpendicular to the YZ-domain, horizontal magnetic field H_y=1/(i*omega*mu) * E_{x,z} with complex unit i and permeability mu. The weighting factor w^s is set to

  • w^s(X) = w_0^s *

if length(X-X^s) <= eta and zero otherwise. X^s is the location of impedance measurement Z_XY^s, w_0^s is the level of confidence (eg. 1/measurement error) and eta is level of spatial confidence.

H_x is given as solution of the PDE

  • -(rho*H_{x,i})_{,i} + i omega * mu * H_x = 0

where H_x at top and bottom is set to solution for background field. Homogeneous Neuman conditions are assumed elsewhere.

__init__(domain, omega, x, Z_YX, eta=None, w0=1.0, mu=1.2566370614359173e-06, sigma0=0.01, airLayerLevel=None, coordinates=None, tol=1e-08, saveMemory=False, directSolver=True)

initializes a new forward model. See base class for a description of the arguments.

getArguments(rho)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

rho (Data of shape (2,)) – resistivity

Returns

Hx, grad(Hx)

Return type

tuple of Data

getDefect(rho, Hx, g_Hx)

Returns the defect value.

Parameters

  • rho (Data of shape ()) – a suggestion for resistivity

  • Hx (Data of shape (2,)) – magnetic field

  • g_Hx (Data of shape (2,2)) – gradient of magnetic field

Return type

float

getGradient(rho, Hx, g_Hx)

Returns the gradient of the defect with respect to resistivity.

Parameters

  • rho (Data of shape ()) – a suggestion for resistivity

  • Hx (Data of shape (2,)) – magnetic field

  • g_Hx (Data of shape (2,2)) – gradient of magnetic field

class esys.downunder.MTMapping(sigma_prior, a=1.0)

mt mapping

sigma=sigma0*exp(a*m)

__init__(sigma_prior, a=1.0)

initializes the mapping

Parameters

sigma_prior – a a-priori conductivity

getDerivative(m)

returns the derivative of the mapping for m

getInverse(s)

returns the value of the inverse of the mapping for s

getValue(m)

returns the value of the mapping for m

class esys.downunder.MagneticIntensityModel(domain, w, b, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Forward Model for magnetic intensity inversion as described in the inversion cookbook.

__init__(domain, w, b, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Creates a new magnetic intensity model on the given domain with one or more surveys (w, b).

Parameters

  • domain (Domain) – domain of the model

  • w (Scalar or list of Scalar) – data weighting factors

  • b (Scalar or list of Scalar) – magnetic intensity field data

  • tol (positive float) – tolerance of underlying PDE

  • background_magnetic_flux_density (Vector or list of float) – background magnetic flux density (in Tesla) with components (B_east, B_north, B_vertical)

  • coordinates (None) – defines coordinate system to be used

  • fixPotentialAtBottom (bool) – if true potential is fixed to zero at the bottom of the domain in addition to the top

getArguments(k)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

k (Scalar) – susceptibility

Returns

scalar magnetic potential and corresponding magnetic field

Return type

Scalar, Vector

getDefect(k, phi, magnetic_flux_density)

Returns the value of the defect.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

float

getGradient(k, phi, magnetic_flux_density)

Returns the gradient of the defect with respect to susceptibility.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

Scalar

getPotential(k)

Calculates the magnetic potential for a given susceptibility.

Parameters

k (Scalar) – susceptibility

Returns

magnetic potential

Return type

Scalar

rescaleWeights(scale=1.0, k_scale=1.0)

rescales the weights such that

sum_s integrate( ( w_i[s] *B_i[s]) (w_j[s]*1/L_j) * L**2 * (background_magnetic_flux_density_j[s]*1/L_j) * k_scale )=scale

Parameters

  • scale (positive float) – scale of data weighting factors

  • k_scale (Scalar) – scale of susceptibility.

class esys.downunder.MagneticInversion(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

Driver class to perform an inversion of magnetic anomaly data. The class uses the standard Regularization class for a single level set function, SusceptibilityMapping mapping and the linear magnetic forward model MagneticModel.

__init__(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

creates an instance of an inversion driver.

Parameters

  • solverclass ('AbstractMinimizer'.) – class of the solver to be used.

  • self_demagnetization – if True self-demagnitization is applied.

  • magnetic_intensity_data – if True magnetic intensity is used in the cost function.

setInitialGuess(k=None)

set the initial guess k for susceptibility for the inversion iteration. If no k present then an appropriate initial guess is chosen.

Parameters

k (Scalar) – initial value for the susceptibility anomaly.

setup(domainbuilder, k0=None, dk=None, z0=None, beta=None, w0=None, w1=None, k_at_depth=None)

Sets up the inversion from a DomainBuilder. If magnetic data are given as scalar it is assumed that values are collected in direction of the background magnetic field.

Parameters

  • domainbuilder (DomainBuilder) – Domain builder object with gravity source(s)

  • k0 (float or Scalar) – reference susceptibility, see SusceptibilityMapping. If not specified, zero is used.

  • dk (float or Scalar) – susceptibility scale, see SusceptibilityMapping. If not specified, 1. is used.

  • z0 (float or Scalar) – reference depth for depth weighting, see SusceptibilityMapping. If not specified, zero is used.

  • beta (float or Scalar) – exponent for depth weighting, see SusceptibilityMapping. If not specified, zero is used.

  • w0 (Scalar or float) – weighting factor for level set term regularization. If not set zero is assumed.

  • w1 (Vector or list of float) – weighting factor for the gradient term in the regularization. If not set zero is assumed

  • k_at_depth (float or None) – value for susceptibility at depth, see DomainBuilder.

siloWriterCallback(k, x, Jx, g_Jx, norm_dJ=None, norm_dx=None)

callback function that can be used to track the solution

Parameters

  • k – iteration count

  • x – current approximation

  • Jx – value of cost function

  • g_Jx – gradient of f at x

class esys.downunder.MagneticModel(domain, w, B, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Forward Model for magnetic inversion as described in the inversion cookbook.

__init__(domain, w, B, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Creates a new magnetic model on the given domain with one or more surveys (w, B).

Parameters

  • domain (Domain) – domain of the model

  • w (Vector or list of Vector) – data weighting factors

  • B (Vector or list of Vector) – magnetic field data

  • tol (positive float) – tolerance of underlying PDE

  • background_magnetic_flux_density (Vector or list of float) – background magnetic flux density (in Tesla) with components (B_east, B_north, B_vertical)

  • coordinates (ReferenceSystem or SpatialCoordinateTransformation) – defines coordinate system to be used

  • fixPotentialAtBottom (bool) – if true potential is fixed to zero at the bottom of the domain in addition to the top

getArguments(k)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

k (Scalar) – susceptibility

Returns

scalar magnetic potential and corresponding magnetic field

Return type

Scalar, Vector

getDefect(k, phi, magnetic_flux_density)

Returns the value of the defect.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

float

getGradient(k, phi, magnetic_flux_density)

Returns the gradient of the defect with respect to susceptibility.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

Scalar

getPotential(k)

Calculates the magnetic potential for a given susceptibility.

Parameters

k (Scalar) – susceptibility

Returns

magnetic potential

Return type

Scalar

rescaleWeights(scale=1.0, k_scale=1.0)

rescales the weights such that

sum_s integrate( ( w_i[s] *B_i[s]) (w_j[s]*1/L_j) * L**2 * (background_magnetic_flux_density_j[s]*1/L_j) * k_scale )=scale

Parameters

  • scale (positive float) – scale of data weighting factors

  • k_scale (Scalar) – scale of susceptibility.

class esys.downunder.Mapping(*args)

An abstract mapping class to map level set functions m to physical parameters p.

__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

getDerivative(m)

returns the value for the derivative of the mapping for m

getInverse(s)

returns the value of the inverse of the mapping for physical parameter p

getTypicalDerivative()

returns a typical value for the derivative

getValue(m)

returns the value of the mapping for m

class esys.downunder.MeteredCostFunction

This an intrumented version of the CostFunction class. The function calls update statistical information. The actual work is done by the methods with corresponding name and a leading underscore. These functions need to be overwritten for a particular cost function implementation.

__init__()

the base constructor initializes the counters so subclasses should ensure the super class constructor is called.

getArguments(x)

returns precalculated values that are shared in the calculation of f(x) and grad f(x) and the Hessian operator

Note

The tuple returned by this call will be passed back to this CostFunction in other calls(eg: getGradient). Its contents are not specified at this level because no code, other than the CostFunction which created it, will be interacting with it. That is, the implementor can put whatever information they find useful in it.

Parameters

x (x-type) – location of derivative

Return type

tuple

getDualProduct(x, r)

returns the dual product of x and r

Return type

float

getGradient(x, *args)

returns the gradient of f at x using the precalculated values for x.

Parameters

  • x (x-type) – location of derivative

  • args – pre-calculated values for x from getArguments()

Return type

r-type

getInverseHessianApproximation(x, r, *args)

returns an approximative evaluation p of the inverse of the Hessian operator of the cost function for a given gradient r at a given location x: H(x) p = r

Note

In general it is assumed that the Hessian H(x) needs to be calculate in each call for a new location x. However, the solver may suggest that this is not required, typically when the iteration is close to completeness.

Parameters

  • x (x-type) – location of Hessian operator to be evaluated.

  • r (r-type) – a given gradient

  • args – pre-calculated values for x from getArguments()

Return type

x-type

getNorm(x)

returns the norm of x

Return type

float

getValue(x, *args)

returns the value f(x) using the precalculated values for x.

Parameters

x (x-type) – a solution approximation

Return type

float

resetCounters()

resets all statistical counters

class esys.downunder.MinimizerException

This is a generic exception thrown by a minimizer.

__init__(*args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

class esys.downunder.MinimizerIterationIncurableBreakDown

Exception thrown if the iteration scheme encountered an incurable breakdown.

__init__(*args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

class esys.downunder.MinimizerLBFGS(J=None, m_tol=0.0001, J_tol=None, imax=300)

Minimizer that uses the limited-memory Broyden-Fletcher-Goldfarb-Shanno method. See Chapter 6 of ‘Numerical Optimization’ by J. Nocedal for an explanation.

__init__(J=None, m_tol=0.0001, J_tol=None, imax=300)

Initializes a new minimizer for a given cost function.

Parameters

  • J (CostFunction) – the cost function to be minimized

  • m_tol (float) – terminate interations when relative change of the level set function is less than or equal m_tol

getOptions()

returns a dictionary of LBFGS options rtype: dictionary with keys ‘truncation’, ‘initialHessian’, ‘restart’, ‘max_linesearch_steps’, ‘max_zoom_steps’ ‘interpolationOrder’, ‘tol_df’, ‘tol_sm’

run(x)
The callback function is called with the following arguments:

k - iteration number x - current estimate Jx - value of cost function at x g_Jx - gradient of cost function at x norm_dJ - ||Jx_k - Jx_{k-1}|| (only if J_tol is set) norm_dx - ||x_k - x_{k-1}|| (only if m_tol is set)

Parameters

x (Data) – Level set function representing our initial guess

Returns

Level set function representing the solution

Return type

Data

setOptions(**opts)

setOptions for LBFGS. use solver.setOptions( key = value) :key truncation: sets the number of previous LBFGS iterations to keep :type truncation : int :default truncation: 30 :key initialHessian: 1 if initial Hessian is given :type initialHessian: 1 or 0 :default initialHessian: 1 :key restart: restart after this many iteration steps :type restart: int :default restart: 60 :key max_linesearch_steps: maximum number of line search iterations :type max_linesearch_steps: int :default max_linesearch_steps: 25 :key max_zoom_steps: maximum number of zoom iterations :type max_zoom_steps: int :default max_zoom_steps: 60 :key interpolationOrder: order of the interpolation used for line search :type interpolationOrder: 1,2,3 or 4 :default interpolationOrder: 1 :key tol_df: interpolated value of alpha, alpha_i, must differ

: from the previous value by at least this much : abs(alpha_i-alpha_{i-1}) < tol_df (line search)

Default tol_df

1e-6

Key tol_sm

interpolated value of alpha must not be less than tol_sm : abs(alpha_i) < tol_sm*abs(alpha_{i-1})

Default tol_sm

1e-5

Example of usage::

cf=DerivedCostFunction() solver=MinimizerLBFGS(J=cf, m_tol = 1e-5, J_tol = 1e-5, imax=300) solver.setOptions(truncation=20, tol_df =1e-7) solver.run(initial_m) result=solver.getResult()

class esys.downunder.MinimizerMaxIterReached

Exception thrown if the maximum number of iteration steps is reached.

__init__(*args, **kwargs)

Initialize self. See help(type(self)) for accurate signature.

class esys.downunder.MinimizerNLCG(J=None, m_tol=0.0001, J_tol=None, imax=300)

Minimizer that uses the nonlinear conjugate gradient method (Fletcher-Reeves variant).

__init__(J=None, m_tol=0.0001, J_tol=None, imax=300)

Initializes a new minimizer for a given cost function.

Parameters

  • J (CostFunction) – the cost function to be minimized

  • m_tol (float) – terminate interations when relative change of the level set function is less than or equal m_tol

run(x)
The callback function is called with the following arguments:

k - iteration number x - current estimate Jx - value of cost function at x g_Jx - gradient of cost function at x gnorm - norm of g_Jx (stopping criterion)

class esys.downunder.NetCdfData(data_type, filename, altitude=0.0, data_variable=None, error=None, scale_factor=None, null_value=None, reference_system=None)

Data Source for gridded netCDF data that use CF/COARDS conventions.

__init__(data_type, filename, altitude=0.0, data_variable=None, error=None, scale_factor=None, null_value=None, reference_system=None)
Parameters

  • filename (str) – file name for survey data in netCDF format

  • data_type (int) – type of data, must be GRAVITY or MAGNETIC

  • altitude (float) – altitude of measurements in meters

  • data_variable (str) – name of the netCDF variable that holds the data. If not provided an attempt is made to determine the variable and an exception thrown on failure.

  • error (str or float) – either the name of the netCDF variable that holds the uncertainties of the measurements or a constant value to use for the uncertainties. If a constant value is supplied, it is scaled by the same factor as the measurements. If not provided the error is assumed to be 2 units for all measurements (i.e. 0.2 mGal and 2 nT for gravity and magnetic, respectively)

  • scale_factor (float) – the measurements and error values are scaled by this factor. By default, gravity data is assumed to be given in 1e-6 m/s^2 (0.1 mGal), while magnetic data is assumed to be in 1e-9 T (1 nT).

  • null_value (float) – value that is used in the file to mark undefined areas. This information is usually included in the file.

  • reference_system (ReferenceSystem) – reference coordinate system to be used. For a Cartesian reference (default) the appropriate UTM transformation is applied.

Note

it is the responsibility of the caller to ensure all data sources and the domain builder use the same reference system.

getDataExtents()

returns ( (x0, y0), (nx, ny), (dx, dy) )

getDataType()

Returns the type of survey data managed by this source. Subclasses must return GRAVITY or MAGNETIC or ACOUSTIC as appropriate.

getScaleFactor()

returns the data scale factor for adjusting measurement to SI

getSurveyData(domain, origin, NE, spacing)

This method is called by the DomainBuilder to retrieve the survey data as Data objects on the given domain.

Subclasses should return one or more Data objects with survey data interpolated on the given escript domain. The exact return type depends on the type of data.

Parameters

  • domain (esys.escript.Domain) – the escript domain to use

  • origin (tuple or list) – the origin coordinates of the domain

  • NE (tuple or list) – the number of domain elements in each dimension

  • spacing (tuple or list) – the cell sizes (node spacing) in the domain

getUtmZone()

All data source coordinates are converted to UTM (Universal Transverse Mercator) in order to have useful domain extents. Subclasses should implement this method and return the UTM zone number of the projected coordinates.

class esys.downunder.NumpyData(data_type, data, error=1.0, length=1000.0, null_value=- 1.0, tags=[], origin=None)
__init__(data_type, data, error=1.0, length=1000.0, null_value=- 1.0, tags=[], origin=None)

A data source that uses survey data from a numpy object or list instead of a file. The dimensionality is inferred from the shape of data (1- and 2-dimensional data is supported). The data origin is assumed to be at the coordinate origin.

Parameters

  • data_type (DataSource.GRAVITY, DataSource.MAGNETIC) – data type indicator

  • data (numpy.array or list) – the survey data array. Note that for a cartesian coordinate system the shape of the data is considered to be (nz,ny,nx).

  • error (float or list or ndarray) – constant value to use for the data uncertainties or a numpy object with uncertainties for every data point.

  • length (float or list or ndarray) – side length(s) of the data slice/volume. This can be a scalar to indicate constant length in all dimensions or an array/list of values in each coordinate dimension.

  • null_value (float) – value that is used in the undefined regions of the survey data object.

  • tags (list of almost any type (typically str)) – a list of tags associated with the data set.

  • origin (list of float) – offset of origin of offset

getDataExtents()

returns a tuple of tuples ( (x0, y0), (nx, ny), (dx, dy) ), where

  • x0, y0 = coordinates of data origin

  • nx, ny = number of data points in x and y

  • dx, dy = spacing of data points in x and y

This method must be implemented in subclasses.

getDataType()

Returns the type of survey data managed by this source. Subclasses must return GRAVITY or MAGNETIC or ACOUSTIC as appropriate.

getSurveyData(domain, origin, NE, spacing)

This method is called by the DomainBuilder to retrieve the survey data as Data objects on the given domain.

Subclasses should return one or more Data objects with survey data interpolated on the given escript domain. The exact return type depends on the type of data.

Parameters

  • domain (esys.escript.Domain) – the escript domain to use

  • origin (tuple or list) – the origin coordinates of the domain

  • NE (tuple or list) – the number of domain elements in each dimension

  • spacing (tuple or list) – the cell sizes (node spacing) in the domain

getUtmZone()

returns a dummy UTM zone since this class does not use real coordinate values.

class esys.downunder.PoleDipoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

Forward model class for a poledipole setup

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • n (float or int) – multiple of spacing between electrodes. typicaly interger

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey:

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

class esys.downunder.PolePoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)

Forward model class for a polepole setup

__init__(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey:

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

returns a list containing 3 lists one for each the primary, secondary and total potential.

class esys.downunder.ReferenceSystem(name='none')

Generic identifier for coordinate systems.

__init__(name='none')

initialization of reference system

Parameters

name (str) – name of the reference system

createTransformation(domain)

creates an appropriate coordinate transformation on a given domain

Note

needs to be overwritten by a particular reference system

Parameters

domain (esys.escript.AbstractDomain) – domain of transformation

Return type

SpatialCoordinateTransformation

getName()

returns the name of the reference system

isCartesian()

returns if the reference system is Cartesian

Note

needs to be overwritten by a particular reference system

Return type

bool

isTheSame(other)

test if argument other defines the same reference system

Parameters

other (ReferenceSystem) – a second reference system

Returns

True if other defines the same reference system

Return type

bool

Note

needs to be overwritten by a particular reference system

class esys.downunder.Regularization(domain, numLevelSets=1, w0=None, w1=None, wc=None, location_of_set_m=<esys.escriptcore.escriptcpp.Data object>, useDiagonalHessianApproximation=False, tol=1e-08, coordinates=None, scale=None, scale_c=None)

The regularization term for the level set function m within the cost function J for an inversion:

J(m)=1/2 * sum_k integrate( mu[k] * ( w0[k] * m_k**2 * w1[k,i] * m_{k,i}**2) + sum_l<k mu_c[l,k] wc[l,k] * | curl(m_k) x curl(m_l) |^2

where w0[k], w1[k,i] and wc[k,l] are non-negative weighting factors and mu[k] and mu_c[l,k] are trade-off factors which may be altered during the inversion. The weighting factors are normalized such that their integrals over the domain are constant:

integrate(w0[k] + inner(w1[k,:],1/L[:]**2))=scale[k] volume(domain)* integrate(wc[l,k]*1/L**4)=scale_c[k] volume(domain) *

__init__(domain, numLevelSets=1, w0=None, w1=None, wc=None, location_of_set_m=<esys.escriptcore.escriptcpp.Data object>, useDiagonalHessianApproximation=False, tol=1e-08, coordinates=None, scale=None, scale_c=None)

initialization.

Parameters

  • domain (Domain) – domain

  • numLevelSets (int) – number of level sets

  • w0 (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – weighting factor for the m**2 term. If not set zero is assumed.

  • w1 (Vector if numLevelSets == 1 or Data object of shape (numLevelSets , DIM) if numLevelSets > 1) – weighting factor for the grad(m_i) terms. If not set zero is assumed

  • wc (Data object of shape (numLevelSets , numLevelSets)) – weighting factor for the cross gradient terms. If not set zero is assumed. Used for the case if numLevelSets > 1 only. Only values wc[l,k] in the lower triangle (l<k) are used.

  • location_of_set_m (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – marks location of zero values of the level set function m by a positive entry.

  • useDiagonalHessianApproximation (bool) – if True cross gradient terms between level set components are ignored when calculating approximations of the inverse of the Hessian Operator. This can speed-up the calculation of the inverse but may lead to an increase of the number of iteration steps in the inversion.

  • tol (positive float) – tolerance when solving the PDE for the inverse of the Hessian Operator

  • coordinates (ReferenceSystem` or SpatialCoordinateTransformation) – defines coordinate system to be used

  • scale (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – weighting factor for level set function variation terms. If not set one is used.

  • scale_c (Data object of shape (numLevelSets,``numLevelSets``)) – scale for the cross gradient terms. If not set one is assumed. Used for the case if numLevelSets > 1 only. Only values scale_c[l,k] in the lower triangle (l<k) are used.

getArguments(m)
getCoordinateTransformation()

returns the coordinate transformation being used

Return type

CoordinateTransformation

getDomain()

returns the domain of the regularization term

Return type

Domain

getDualProduct(m, r)

returns the dual product of a gradient represented by X=r[1] and Y=r[0] with a level set function m:

Y_i*m_i + X_ij*m_{i,j}

Return type

float

getGradient(m, grad_m)

returns the gradient of the cost function J with respect to m.

Note

This implementation returns Y_k=dPsi/dm_k and X_kj=dPsi/dm_kj

getInverseHessianApproximation(m, r, grad_m, solve=True)
getNorm(m)

returns the norm of m.

Parameters

m (Data) – level set function

Return type

float

getNumLevelSets()

returns the number of level set functions

Return type

int

getNumTradeOffFactors()

returns the number of trade-off factors being used.

Return type

int

getPDE()

returns the linear PDE to be solved for the Hessian Operator inverse

Return type

LinearPDE

getValue(m, grad_m)

returns the value of the cost function J with respect to m. This equation is specified in the inversion cookbook.

Return type

float

setTradeOffFactors(mu=None)

sets the trade-off factors for the level-set variation and the cross-gradient.

Parameters

mu (list of float or `numpy.array`) – new values for the trade-off factors where values mu[:numLevelSets] are the trade-off factors for the level-set variation and the remaining values for the cross-gradient part with mu_c[l,k]=mu[numLevelSets+l+((k-1)*k)/2] (l<k). If no values for mu are given ones are used. Values must be positive.

setTradeOffFactorsForCrossGradient(mu_c=None)

sets the trade-off factors for the cross-gradient terms.

Parameters

mu_c (float, list of float or numpy.array) – new values for the trade-off factors for the cross-gradient terms. Values must be positive. If no value is given ones are used. Only value mu_c[l,k] for l<k are used.

setTradeOffFactorsForVariation(mu=None)

sets the trade-off factors for the level-set variation part.

Parameters

mu (float, list of float or `numpy.array`) – new values for the trade-off factors. Values must be positive.

updateHessian()

notifies the class to recalculate the Hessian operator.

class esys.downunder.Ricker(f_dom=40, t_dom=None)

The Ricker Wavelet w=f(t)

__init__(f_dom=40, t_dom=None)

Sets up a Ricker wavelet wih dominant frequence f_dom and center at time t_dom. If t_dom is not given an estimate for suitable t_dom is calculated so f(0)~0.

Note

maximum frequence is about 2 x the dominant frequence.

getAcceleration(t)

get the acceleration f’’(t) at time t

getCenter()

Return value of wavelet center

getTimeScale()

Returns the time scale which is the inverse of the largest frequence with a significant spectral component.

getValue(t)

get value of wavelet at time t

getVelocity(t)

get the velocity f’(t) at time t

class esys.downunder.SchlumbergerSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

Schlumberger survey forward calculation

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

This is a skeleton class for all the other forward modeling classes.

getApparentResistivity(delPhiList)
getElectrodeDict()

retuns the electrode dictionary

getPotentialAnalytic()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

getPotentialNumeric()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

getSourcesSamples()

return a list of tuples of sample locations followed by a list of tuples of source locations.

class esys.downunder.SeismicSource(x, y=0.0, omega=0.0, elevation=0.0, power=None, orientation=None)

describes a seimic source by location (x,y), frequency omega, power (if known) and orientation (if known). this class is used to tag seismic data sources.

__init__(x, y=0.0, omega=0.0, elevation=0.0, power=None, orientation=None)

initiale the source

Parameters

  • x (float) – lateral x location

  • y (float) – lateral y location

  • omega (float) – frequency of source

  • elevation – elevation of source above reference level

  • power (complex or None) – power of source at frequence

  • orientation (vector of appropriate length or None) – oriententation of source in 3D or 2D (or None)

getElevation()

return elevation of source :rtype: float

getFrequency()

return frequency of source :rtype: float

getLocation()

return location of source :rtype: tuple of float

getOrientation()

return power of source orientation at frequency :rtype: vector type object or None

getPower()

return power of source at frequency :rtype: complex or None

class esys.downunder.SelfDemagnetizationModel(domain, w, B, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Forward Model for magnetic inversion with self-demagnetization as described in the inversion cookbook.

__init__(domain, w, B, background_magnetic_flux_density, coordinates=None, fixPotentialAtBottom=False, tol=1e-08)

Creates a new magnetic model on the given domain with one or more surveys (w, B).

Parameters

  • domain (Domain) – domain of the model

  • w (Vector or list of Vector) – data weighting factors

  • B (Vector or list of Vector) – magnetic field data

  • background_magnetic_flux_density (Vector or list of float) – background magnetic flux density (in Tesla) with components (B_east, B_north, B_vertical)

  • coordinates (ReferenceSystem or SpatialCoordinateTransformation) – defines coordinate system to be used

  • fixPotentialAtBottom (bool) – if true potential is fixed to zero at the bottom of the domain in addition to the top

  • tol (positive float) – tolerance of underlying PDE

getArguments(k)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

k (Scalar) – susceptibility

Returns

scalar magnetic potential and corresponding magnetic field

Return type

Scalar, Vector

getDefect(k, phi, grad_phi, magnetic_flux_density)

Returns the value of the defect.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

float

getGradient(k, phi, grad_phi, magnetic_flux_density)

Returns the gradient of the defect with respect to susceptibility.

Parameters

  • k (Scalar) – susceptibility

  • phi (Scalar) – corresponding potential

  • magnetic_flux_density (Vector) – magnetic field

Return type

Scalar

getPotential(k)

Calculates the magnetic potential for a given susceptibility.

Parameters

k (Scalar) – susceptibility

Returns

magnetic potential

Return type

Scalar

rescaleWeights(scale=1.0, k_scale=1.0)

rescales the weights such that

sum_s integrate( ( w_i[s] *B_i[s]) (w_j[s]*1/L_j) * L**2 * (background_magnetic_flux_density_j[s]*1/L_j) * k_scale )=scale

Parameters

  • scale (positive float) – scale of data weighting factors

  • k_scale (Scalar) – scale of susceptibility.

class esys.downunder.SimpleSEGYWriter(receiver_group=None, source=0.0, sampling_interval=0.004, text='some seimic data')

A simple writer for 2D and 3D seismic lines, in particular for synthetic data

Typical usage:

from esys.escript import unitsSI as U
sw=SimpleSEGYWriter([0.,100*U.m,200*U,m,300.], source=200*U.m, sampling_interval=4*U.msec)
while n < 10:
    sw.addRecord([i*2., i*0.67, i**2, -i*7])
sw.write('example.segy')
Note

the writer uses obspy

__init__(receiver_group=None, source=0.0, sampling_interval=0.004, text='some seimic data')

initalize writer

Parameters

  • receiver_group – list of receiver coordinates (in meters). For the 2D case a list of floats is given, for the 3D case a list of coordinate tuples are given

  • source – coordinates of the source (in meters). For the 2D case a single floats is given, for the 3D case a coordinate tuples

  • sampling_interval – sample rate in seconds

  • text – a text for the header file (e.g a description)

COORDINATE_SCALE = 1000.0
addRecord(record)

Adds a record to the traces. A time difference of sample_interval between two records is assumed. The record mast be a list of as many values as given receivers or a float if a single receiver is used.

Parameters

record – list of tracks to be added to the record.

getSamplingInterval()

returns the sampling interval in seconds.

obspy_available()

for checking if the obspy module is available

write(filename)

writes to segy file

Parameters

filename – file name

Note

the function uses the obspy module.

class esys.downunder.SmoothAnomaly(lx, ly, lz, x, y, depth, v_inner=None, v_outer=None)

A source feature in the form of a blob (roughly gaussian).

__init__(lx, ly, lz, x, y, depth, v_inner=None, v_outer=None)

Intializes the smooth anomaly data.

Parameters

  • lx – size of blob in x-dimension

  • ly – size of blob in y-dimension

  • lz – size of blob in z-dimension

  • x – location of blob in x-dimension

  • y – location of blob in y-dimension

  • depth – depth of blob

  • v_inner – value in the centre of the blob

  • v_outer – value in the periphery of the blob

getMask(x)

Returns the mask of the area of interest for this feature. That is, mask is non-zero where the value returned by getValue() should be applied, zero elsewhere.

getValue(x)

Returns the value for the area covered by mask. It can be constant or a Data object with spatial dependency.

class esys.downunder.SonicHTIWave(domain, v_p, wavelet, source_tag, source_vector=[1.0, 0.0], eps=0.0, delta=0.0, azimuth=0.0, dt=None, p0=None, v0=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

Solving the HTI wave equation (along the x_0 axis) with azimuth (rotation around verticle axis) under the assumption of zero shear wave velocities The unknowns are the transversal (along x_0) and vertial stress (Q, P)

Note

In case of a two dimensional domain the second spatial dimenion is depth.

__init__(domain, v_p, wavelet, source_tag, source_vector=[1.0, 0.0], eps=0.0, delta=0.0, azimuth=0.0, dt=None, p0=None, v0=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

initialize the HTI wave solver

Parameters

  • domain (Doamin) – domain of the problem

  • v_p (escript.Scalar) – vertical p-velocity field

  • v_s (escript.Scalar) – vertical s-velocity field

  • wavelet (Wavelet) – wavelet to describe the time evolution of source term

  • source_tag ('str' or 'int') – tag of the source location

  • source_vector – source orientation vector

  • eps – first Thompsen parameter

  • azimuth – azimuth (rotation around verticle axis)

  • gamma – third Thompsen parameter

  • rho – density

  • dt – time step size. If not present a suitable time step size is calculated.

  • p0 – initial solution (Q(t=0), P(t=0)). If not present zero is used.

  • v0 – initial solution change rate. If not present zero is used.

  • absorption_zone – thickness of absorption zone

  • absorption_cut – boundary value of absorption decay factor

  • lumping – if True mass matrix lumping is being used. This is accelerates the computing but introduces some diffusion.

class esys.downunder.SonicWave(domain, v_p, wavelet, source_tag, dt=None, p0=None, p0_t=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

Solving the sonic wave equation

p_tt = (v_p**2 * p_i)_i  + f(t) * delta_s where (p-) velocity v_p.

f(t) is wavelet acting at a point source term at positon s

__init__(domain, v_p, wavelet, source_tag, dt=None, p0=None, p0_t=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

initialize the sonic wave solver

Parameters

  • domain (Domain) – domain of the problem

  • v_p (escript.Scalar) – p-velocity field

  • wavelet (Wavelet) – wavelet to describe the time evolution of source term

  • source_tag ('str' or 'int') – tag of the source location

  • dt – time step size. If not present a suitable time step size is calculated.

  • p0 – initial solution. If not present zero is used.

  • p0_t – initial solution change rate. If not present zero is used.

  • absorption_zone – thickness of absorption zone

  • absorption_cut – boundary value of absorption decay factor

  • lumping – if True mass matrix lumping is being used. This is accelerates the computing but introduces some diffusion.

class esys.downunder.SpatialCoordinateTransformation(domain, reference=<esys.downunder.coordinates.CartesianReferenceSystem object>)

Defines an orthogonal coordinate transformation from a domain into the Cartesian domain using a coordinate transformation.

The default implementation is the identity transformation (i.e. no changes are applied to the domain). Overwrite the appropriate methods to define other coordinate systems.

__init__(domain, reference=<esys.downunder.coordinates.CartesianReferenceSystem object>)

set up the orthogonal coordinate transformation.

Parameters

  • domain (esys.escript.AbstractDomain) – domain in the domain of the coordinate transformation

  • reference (ReferenceSystem) – the reference system

getDomain()

returns the domain of the coordinate transformation.

Return type

esys.escript.AbstractDomain

getGradient(u)

returns the gradient of a scalar function in direction of the coordinate axis.

Return type

esys.escript.Vector

getReferenceSystem()

returns the reference system used to to define the coordinate transformation

Return type

ReferenceSystem

getScalingFactors()

returns the scaling factors

Return type

esys.escript.Vector

getVolumeFactor()

returns the volume factor for the coordinate transformation

Return type

esys.escript.Scalar

isCartesian()

returns True if the scaling factors (and the volume factor) are equal to 1

Return type

bool

isTheSame(other)

test if argument other defines the same coordinate transformation

Parameters

other (SpatialCoordinateTransformation) – a second coordinate transformation

Returns

True if other defines then same coordinate transformation

Return type

bool

class esys.downunder.SplitInversionCostFunction(numLevelSets=None, numModels=None, numMappings=None, splitworld=None, worldsinit_fn=None)

Class to define cost function J(m) for inversion with one or more forward models based on a multi-valued level set function m:

J(m) = J_reg(m) + sum_f mu_f * J_f(p)

where J_reg(m) is the regularization and cross gradient component of the cost function applied to a level set function m, J_f(p) are the data defect cost functions involving a physical forward model using the physical parameter(s) p and mu_f is the trade-off factor for model f.

A forward model depends on a set of physical parameters p which are constructed from components of the level set function m via mappings.

Example 1 (single forward model):

m=Mapping() f=ForwardModel() J=InversionCostFunction(Regularization(), m, f)

Example 2 (two forward models on a single valued level set)

m0=Mapping() m1=Mapping() f0=ForwardModel() f1=ForwardModel()

J=InversionCostFunction(Regularization(), mappings=[m0, m1], forward_models=[(f0, 0), (f1,1)])

Example 3 (two forward models on 2-valued level set)

m0=Mapping() m1=Mapping() f0=ForwardModel() f1=ForwardModel()

J=InversionCostFunction(Regularization(self.numLevelSets=2), mappings=[(m0,0), (m1,0)], forward_models=[(f0, 0), (f1,1)])

Note

If provides_inverse_Hessian_approximation is true, then the class provides an approximative inverse of the Hessian operator.

__init__(numLevelSets=None, numModels=None, numMappings=None, splitworld=None, worldsinit_fn=None)

fill this in.

calculateGradient(vnames1, vnames2)

The gradient operation produces two components (designated (Y^,X) in the non-split version). vnames1 gives the variable name(s) where the first component should be stored. vnames2 gives the variable name(s) where the second component should be stored.

static calculatePropertiesHelper(self, m, mappings)

returns a list of the physical properties from a given level set function m using the mappings of the cost function.

Parameters

m (Data) – level set function

Return type

list of Data

calculateValue(vname)
createLevelSetFunction(*props)

returns an instance of an object used to represent a level set function initialized with zeros. Components can be overwritten by physical properties props. If present entries must correspond to the mappings arguments in the constructor. Use None for properties for which no value is given.

static createLevelSetFunctionHelper(self, regularization, mappings, *props)

Returns an object (init-ed) with 0s. Components can be overwritten by physical properties props. If present entries must correspond to the mappings arguments in the constructor. Use None for properties for which no value is given.

static formatMappings(mappings, numLevelSets)
static formatModels(forward_models, numMappings)
getComponentValues(m, *args)
getDomain()

returns the domain of the cost function

Return type

Domain

getForwardModel(idx=None)

returns the idx-th forward model.

Parameters

idx (int) – model index. If cost function contains one model only idx can be omitted.

static getModelArgs(self, fwdmodels)

Attempts to import the arguments for forward models, if they are not available, Computes and exports them

getNumTradeOffFactors()

returns the number of trade-off factors being used including the trade-off factors used in the regularization component.

Return type

int

getProperties(m, return_list=False)

returns a list of the physical properties from a given level set function m using the mappings of the cost function.

Parameters

  • m (Data) – level set function

  • return_list (bool) – if True a list is returned.

Return type

list of Data

getRegularization()

returns the regularization

Return type

Regularization

getTradeOffFactors(mu=None)

returns a list of the trade-off factors.

Return type

list of float

getTradeOffFactorsModels()

returns the trade-off factors for the forward models

Return type

float or list of float

provides_inverse_Hessian_approximation = True
setPoint()
setTradeOffFactors(mu=None)

sets the trade-off factors for the forward model and regularization terms.

Parameters

mu (list of float) – list of trade-off factors.

setTradeOffFactorsModels(mu=None)

sets the trade-off factors for the forward model components.

Parameters

mu (float in case of a single model or a list of float with the length of the number of models.) – list of the trade-off factors. If not present ones are used.

setTradeOffFactorsRegularization(mu=None, mu_c=None)

sets the trade-off factors for the regularization component of the cost function, see Regularization for details.

Parameters

  • mu – trade-off factors for the level-set variation part

  • mu_c – trade-off factors for the cross gradient variation part

static subworld_setMu_model(self, **args)
updateHessian()

notifies the class that the Hessian operator needs to be updated.

static update_point_helper(self, newpoint)

Call within a subworld to set ‘current_point’ to newpoint and update all the cached args info

class esys.downunder.SplitMinimizerLBFGS(J=None, m_tol=0.0001, J_tol=None, imax=300)

Minimizer that uses the limited-memory Broyden-Fletcher-Goldfarb-Shanno method.

version modified to fit with split world.

__init__(J=None, m_tol=0.0001, J_tol=None, imax=300)

Initializes a new minimizer for a given cost function.

Parameters

  • J (CostFunction) – the cost function to be minimized

  • m_tol (float) – terminate interations when relative change of the level set function is less than or equal m_tol

getOptions()

Returns a dictionary of minimizer-specific options.

static move_point_from_base(self, **kwargs)
run()

This version relies on the costfunction already having an initial guess loaded. It also does not return the result, meaning a job needs to be submitted to get the result out.

setOptions(**opts)

Sets minimizer-specific options. For a list of possible options see getOptions().

class esys.downunder.SplitRegularization(domain, numLevelSets=1, w0=None, w1=None, wc=None, location_of_set_m=<esys.escriptcore.escriptcpp.Data object>, useDiagonalHessianApproximation=False, tol=1e-08, coordinates=None, scale=None, scale_c=None)

The regularization term for the level set function m within the cost function J for an inversion:

J(m)=1/2 * sum_k integrate( mu[k] * ( w0[k] * m_k**2 * w1[k,i] * m_{k,i}**2) + sum_l<k mu_c[l,k] wc[l,k] * | curl(m_k) x curl(m_l) |^2

where w0[k], w1[k,i] and wc[k,l] are non-negative weighting factors and mu[k] and mu_c[l,k] are trade-off factors which may be altered during the inversion. The weighting factors are normalized such that their integrals over the domain are constant:

integrate(w0[k] + inner(w1[k,:],1/L[:]**2))=scale[k] volume(domain)* integrate(wc[l,k]*1/L**4)=scale_c[k] volume(domain) *

__init__(domain, numLevelSets=1, w0=None, w1=None, wc=None, location_of_set_m=<esys.escriptcore.escriptcpp.Data object>, useDiagonalHessianApproximation=False, tol=1e-08, coordinates=None, scale=None, scale_c=None)

initialization.

Parameters

  • domain (Domain) – domain

  • numLevelSets (int) – number of level sets

  • w0 (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – weighting factor for the m**2 term. If not set zero is assumed.

  • w1 (Vector if numLevelSets == 1 or Data object of shape (numLevelSets , DIM) if numLevelSets > 1) – weighting factor for the grad(m_i) terms. If not set zero is assumed

  • wc (Data object of shape (numLevelSets , numLevelSets)) – weighting factor for the cross gradient terms. If not set zero is assumed. Used for the case if numLevelSets > 1 only. Only values wc[l,k] in the lower triangle (l<k) are used.

  • location_of_set_m (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – marks location of zero values of the level set function m by a positive entry.

  • useDiagonalHessianApproximation (bool) – if True cross gradient terms between level set components are ignored when calculating approximations of the inverse of the Hessian Operator. This can speed-up the calculation of the inverse but may lead to an increase of the number of iteration steps in the inversion.

  • tol (positive float) – tolerance when solving the PDE for the inverse of the Hessian Operator

  • coordinates (ReferenceSystem` or SpatialCoordinateTransformation) – defines coordinate system to be used

  • scale (Scalar if numLevelSets == 1 or Data object of shape (numLevelSets ,) if numLevelSets > 1) – weighting factor for level set function variation terms. If not set one is used.

  • scale_c (Data object of shape (numLevelSets,``numLevelSets``)) – scale for the cross gradient terms. If not set one is assumed. Used for the case if numLevelSets > 1 only. Only values scale_c[l,k] in the lower triangle (l<k) are used.

getArguments(m)
getCoordinateTransformation()

returns the coordinate transformation being used

Return type

CoordinateTransformation

getDomain()

returns the domain of the regularization term

Return type

Domain

getDualProduct(m, r)

returns the dual product of a gradient represented by X=r[1] and Y=r[0] with a level set function m:

Y_i*m_i + X_ij*m_{i,j}

Return type

float

getGradient()

returns the gradient of f at x using the precalculated values for x.

Parameters

  • x (x-type) – location of derivative

  • args – pre-calculated values for x from getArguments()

Return type

r-type

getGradientAtPoint()

returns the gradient of the cost function J with respect to m.

Note

This implementation returns Y_k=dPsi/dm_k and X_kj=dPsi/dm_kj

getInverseHessianApproximationAtPoint(r, solve=True)
getNorm(m)

returns the norm of m.

Parameters

m (Data) – level set function

Return type

float

getNumLevelSets()

returns the number of level set functions

Return type

int

getNumTradeOffFactors()

returns the number of trade-off factors being used.

Return type

int

getPDE()

returns the linear PDE to be solved for the Hessian Operator inverse

Return type

linearPDEs.LinearPDE

getValue(m, grad_m)

returns the value of the cost function J with respect to m. This equation is specified in the inversion cookbook.

Return type

float

getValueAtPoint()

returns the value of the cost function J with respect to m. This equation is specified in the inversion cookbook.

Return type

float

setPoint(m)

sets the point which this function will work with

Parameters

m (Data) – level set function

setTradeOffFactors(mu=None)

sets the trade-off factors for the level-set variation and the cross-gradient.

Parameters

mu (list of float or `numpy.array`) – new values for the trade-off factors where values mu[:numLevelSets] are the trade-off factors for the level-set variation and the remaining values for the cross-gradient part with mu_c[l,k]=mu[numLevelSets+l+((k-1)*k)/2] (l<k). If no values for mu are given ones are used. Values must be positive.

setTradeOffFactorsForCrossGradient(mu_c=None)

sets the trade-off factors for the cross-gradient terms.

Parameters

mu_c (float, list of float or numpy.array) – new values for the trade-off factors for the cross-gradient terms. Values must be positive. If no value is given ones are used. Only value mu_c[l,k] for l<k are used.

setTradeOffFactorsForVariation(mu=None)

sets the trade-off factors for the level-set variation part.

Parameters

mu (float, list of float or `numpy.array`) – new values for the trade-off factors. Values must be positive.

updateHessian()

notifies the class to recalculate the Hessian operator.

class esys.downunder.StrongJointGravityMagneticInversion(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

Driver class to perform a joint inversion of Gravity (Bouguer) and magnetic anomaly data with the assumption that there is a functional relationship between density and susceptibility.

The class uses the standard Regularization class for a single level set function, DensityMapping and SusceptibilityMapping mappings, the gravity forward model GravityModel and the linear magnetic forward model MagneticModel.

__init__(solverclass=None, debug=False, self_demagnetization=False, magnetic_intensity_data=False)

creates an instance of an inversion driver.

Parameters

  • solverclass ('AbstractMinimizer'.) – class of the solver to be used.

  • self_demagnetization – if True self-demagnitization is applied.

  • magnetic_intensity_data – if True magnetic intensity is used in the cost function.

DENSITY = 0
SUSCEPTIBILITY = 1
setInitialGuess(rho=None, k=None)

set the initial guess rho for density and k for susceptibility for the inversion iteration.

Parameters

  • rho (Scalar) – initial value for the density anomaly.

  • k (Scalar) – initial value for the susceptibility anomaly.

setup(domainbuilder, rho0=None, drho=None, rho_z0=None, rho_beta=None, k0=None, dk=None, k_z0=None, k_beta=None, w0=None, w1=None, w_gc=None, rho_at_depth=None, k_at_depth=None)

Sets up the inversion from an instance domainbuilder of a DomainBuilder. Gravity and magnetic data attached to the domainbuilder are considered in the inversion. If magnetic data are given as scalar it is assumed that values are collected in direction of the background magnetic field.

Parameters

  • domainbuilder (DomainBuilder) – Domain builder object with gravity source(s)

  • rho0 (float or Scalar) – reference density, see DensityMapping. If not specified, zero is used.

  • drho (float or Scalar) – density scale, see DensityMapping. If not specified, 2750 kg/m^3 is used.

  • rho_z0 (float or Scalar) – reference depth for depth weighting for density, see DensityMapping. If not specified, zero is used.

  • rho_beta (float or Scalar) – exponent for density depth weighting, see DensityMapping. If not specified, zero is used.

  • k0 (float or Scalar) – reference susceptibility, see SusceptibilityMapping. If not specified, zero is used.

  • dk (float or Scalar) – susceptibility scale, see SusceptibilityMapping. If not specified, 1. is used.

  • k_z0 (float or Scalar) – reference depth for susceptibility depth weighting, see SusceptibilityMapping. If not specified, zero is used.

  • k_beta (float or Scalar) – exponent for susceptibility depth weighting, see SusceptibilityMapping. If not specified, zero is used.

  • w0 (Scalar or float) – weighting factor for level set term in the regularization. If not set zero is assumed.

  • w1 (es.Data or ndarray of shape (DIM,)) – weighting factor for the gradient term in the regularization see Regularization. If not set zero is assumed.

  • w_gc (Scalar or float) – weighting factor for the cross gradient term in the regularization, see Regularization. If not set one is assumed.

  • k_at_depth (float or None) – value for susceptibility at depth, see DomainBuilder.

  • rho_at_depth (float or None) – value for density at depth, see DomainBuilder.

siloWriterCallback(k, x, Jx, g_Jx, norm_dJ=None, norm_dx=None)

callback function that can be used to track the solution

Parameters

  • k – iteration count

  • x – current m approximation

  • Jx – value of cost function

  • g_Jx – gradient of f at x

class esys.downunder.Subsidence(domain, w, d, lam, mu, coordinates=None, tol=1e-08)

Forward Model for subsidence inversion minimizing integrate( (inner(w,u)-d)**2) where u is the surface displacement due to a pressure change P

__init__(domain, w, d, lam, mu, coordinates=None, tol=1e-08)

Creates a new subsidence on the given domain

Parameters

  • domain (Domain) – domain of the model

  • w (Vector with FunctionOnBoundary) – data weighting factors and direction

  • d (Scalar with FunctionOnBoundary) – displacement measured at surface

  • lam (Scalar with Function) – 1st Lame coefficient

  • lam – 2st Lame coefficient/Shear modulus

  • coordinates (ReferenceSystem or SpatialCoordinateTransformation) – defines coordinate system to be used (not supported yet))

  • tol (positive float) – tolerance of underlying PDE

getArguments(P)

Returns precomputed values shared by getDefect() and getGradient().

Parameters

P (Scalar) – pressure

Returns

displacement u

Return type

Vector

getDefect(P, u)

Returns the value of the defect.

Parameters

  • P (Scalar) – pressure

  • u (Vector) – corresponding displacement

Return type

float

getGradient(P, u)

Returns the gradient of the defect with respect to susceptibility.

Parameters

  • P (Scalar) – pressure

  • u (Vector) – corresponding displacement

Return type

Scalar

rescaleWeights(scale=1.0, P_scale=1.0)

rescales the weights

Parameters

  • scale (positive float) – scale of data weighting factors

  • P_scale (Scalar) – scale of pressure increment

class esys.downunder.SusceptibilityMapping(domain, z0=None, k0=None, dk=None, beta=None)

Susceptibility mapping with depth weighting

k = k0 + dk * ( (x_2 - z0)/l_z)^(beta/2) ) * m

__init__(domain, z0=None, k0=None, dk=None, beta=None)

set up mapping

Parameters

  • domain (Domain) – domain of the mapping

  • z0 (scalar) – depth weighting offset. If not present no depth scaling is applied.

  • k0 (scalar) – reference density, defaults to 0

  • dk (scalar) – susceptibility scale, defaults to 1

  • beta (float) – depth weighting exponent, defaults to 2

class esys.downunder.SyntheticData(data_type, n_length=1, n_depth=1, depth_offset=0.0, depth=None, amplitude=None, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False, s=0.0)

Defines synthetic gravity/magnetic data based on harmonic property anomaly

rho = amplitude * sin(n_depth * pi /depth * (z+depth_offset)) * sin(n_length * pi /length * (x - shift) )

for all x and z<=0. For z>0 rho = 0.

__init__(data_type, n_length=1, n_depth=1, depth_offset=0.0, depth=None, amplitude=None, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False, s=0.0)
Parameters

  • data_type (DataSource.GRAVITY, DataSource.MAGNETIC) – data type indicator

  • n_length (int) – number of oscillations in the anomaly data within the observation region

  • n_depth (int) – number of oscillations in the anomaly data below surface

  • depth_offset (float) – vertical offset of the data

  • depth (float) – vertical extent in the anomaly data. If not present the depth of the domain is used.

  • amplitude – data amplitude. Default value is 200 U.kg/U.m**3 for gravity and 0.1 for magnetic data.

  • DIM (int (2 or 3)) – spatial dimensionality

  • number_of_elements (int) – lateral number of elements in the region where data are collected

  • length (float) – lateral extent of the region where data are collected

  • B_b (list of Scalar) – background magnetic flux density [B_r, B_latiude, B_longitude]. Only used for magnetic data.

  • data_offset (float) – offset of the data collection region from the surface

  • full_knowledge (Bool) – if True data are collected from the entire subsurface region. This is mainly for testing.

getReferenceProperty(domain=None)

Returns the reference Data object that was used to generate the gravity/susceptibility anomaly data.

class esys.downunder.SyntheticDataBase(data_type, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False)

Base class to define reference data based on a given property distribution (density or susceptibility). Data are collected from a square region of vertical extent length on a grid with number_of_elements cells in each direction.

The synthetic data are constructed by solving the appropriate forward problem. Data can be sampled with an offset from the surface at z=0 or using the entire subsurface region.

__init__(data_type, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False)
Parameters

  • data_type (DataSource.GRAVITY, DataSource.MAGNETIC) – data type indicator

  • DIM (int (2 or 3)) – number of spatial dimensions

  • number_of_elements (int) – lateral number of elements in the region where data are collected

  • length (float) – lateral extent of the region where data are collected

  • B_b (list of Scalar) – background magnetic flux density [B_r, B_latiude, B_longitude]. Only used for magnetic data.

  • data_offset (float) – offset of the data collection region from the surface

  • full_knowledge (Bool) – if True data are collected from the entire subsurface region. This is mainly for testing.

getDataExtents()

returns the lateral data extend of the data set

getDataType()

returns the data type

getReferenceProperty(domain=None)

Returns the reference Data object that was used to generate the gravity/susceptibility anomaly data.

Returns

the density or susceptibility anomaly used to create the survey data

Note

it can be assumed that in the first call the domain argument is present so the actual anomaly data can be created. In subsequent calls this may not be true.

Note

method needs to be overwritten

getSurveyData(domain, origin, number_of_elements, spacing)

returns the survey data placed on a given domain.

Parameters

  • domain (Domain) – domain on which the data are to be placed

  • origin (list of float) – origin of the domain

  • number_of_elements (list of int) – number of elements (or cells) in each spatial direction used to span the domain

  • spacing (list of float) – cell size in each spatial direction

Returns

observed gravity field or magnetic flux density for each cell in the domain and for each cell an indicator 1/0 if the data are valid or not.

Return type

pair of Scalar

getUtmZone()

returns a dummy UTM zone since this class does not use real coordinate values.

class esys.downunder.SyntheticFeatureData(data_type, features, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False)

Uses a list of SourceFeature objects to define synthetic anomaly data.

__init__(data_type, features, DIM=2, number_of_elements=10, length=1000.0, B_b=None, data_offset=0, full_knowledge=False)
Parameters

  • data_type (DataSource.GRAVITY, DataSource.MAGNETIC) – data type indicator

  • features (list of SourceFeature) – list of features. It is recommended that the features are located entirely below the surface.

  • DIM (int (2 or 3)) – spatial dimensionality

  • number_of_elements (int) – lateral number of elements in the region where data are collected

  • length (float) – lateral extent of the region where data are collected

  • B_b (list of Scalar) – background magnetic flux density [B_r, B_latiude, B_longitude]. Only used for magnetic data.

  • data_offset (float) – offset of the data collection region from the surface

  • full_knowledge (Bool) – if True data are collected from the entire subsurface region. This is mainly for testing.

getReferenceProperty(domain=None)

Returns the reference Data object that was used to generate the gravity/susceptibility anomaly data.

class esys.downunder.TTIWave(domain, v_p, v_s, wavelet, source_tag, source_vector=[0.0, 1.0], eps=0.0, delta=0.0, theta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

Solving the 2D TTI wave equation with

sigma_xx= c11*e_xx + c13*e_zz + c15*e_xz sigma_zz= c13*e_xx + c33*e_zz + c35*e_xz sigma_xz= c15*e_xx + c35*e_zz + c55*e_xz

the coefficients c11, c13, etc are calculated from the tompsen parameters eps, delta and the tilt theta

Note

currently only the 2D case is supported.

__init__(domain, v_p, v_s, wavelet, source_tag, source_vector=[0.0, 1.0], eps=0.0, delta=0.0, theta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=300.0, absorption_cut=0.01, lumping=True)

initialize the TTI wave solver

Parameters

  • domain (Domain) – domain of the problem

  • v_p (escript.Scalar) – vertical p-velocity field

  • v_s (escript.Scalar) – vertical s-velocity field

  • wavelet (Wavelet) – wavelet to describe the time evolution of source term

  • source_tag ('str' or 'int') – tag of the source location

  • source_vector – source orientation vector

  • eps – first Thompsen parameter

  • delta – second Thompsen parameter

  • theta – tilting (in Rad)

  • rho – density

  • dt – time step size. If not present a suitable time step size is calculated.

  • u0 – initial solution. If not present zero is used.

  • v0 – initial solution change rate. If not present zero is used.

  • absorption_zone – thickness of absorption zone

  • absorption_cut – boundary value of absorption decay factor

  • lumping – if True mass matrix lumping is being used. This is accelerates the computing but introduces some diffusion.

class esys.downunder.VTIWave(domain, v_p, v_s, wavelet, source_tag, source_vector=[0.0, 0.0, 1.0], eps=0.0, gamma=0.0, delta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=None, absorption_cut=0.01, lumping=True, disable_fast_assemblers=False)

Solving the VTI wave equation

Note

In case of a two dimensional domain the second spatial dimenion is depth.

__init__(domain, v_p, v_s, wavelet, source_tag, source_vector=[0.0, 0.0, 1.0], eps=0.0, gamma=0.0, delta=0.0, rho=1.0, dt=None, u0=None, v0=None, absorption_zone=None, absorption_cut=0.01, lumping=True, disable_fast_assemblers=False)

initialize the VTI wave solver

Parameters

  • domain (Domain) – domain of the problem

  • v_p (escript.Scalar) – vertical p-velocity field

  • v_s (escript.Scalar) – vertical s-velocity field

  • wavelet (Wavelet) – wavelet to describe the time evolution of source term

  • source_tag ('str' or 'int') – tag of the source location

  • source_vector – source orientation vector

  • eps – first Thompsen parameter

  • delta – second Thompsen parameter

  • gamma – third Thompsen parameter

  • rho – density

  • dt – time step size. If not present a suitable time step size is calculated.

  • u0 – initial solution. If not present zero is used.

  • v0 – initial solution change rate. If not present zero is used.

  • absorption_zone – thickness of absorption zone

  • absorption_cut – boundary value of absorption decay factor

  • lumping – if True mass matrix lumping is being used. This is accelerates the computing but introduces some diffusion.

  • disable_fast_assemblers (boolean) – if True, forces use of slower and more general PDE assemblers

setQ(q)

sets the PDE q value

Parameters

q – the value to set

class esys.downunder.WaveBase(dt, u0, v0, t0=0.0)

Base for wave propagation using the Verlet scheme.

u_tt = A(t,u), u(t=t0)=u0, u_t(t=t0)=v0

with a given acceleration force as function of time.

a_n=A(t_{n-1}) v_n=v_{n-1} + dt * a_n u_n=u_{n-1} + dt * v_n

__init__(dt, u0, v0, t0=0.0)

set up the wave base

Parameters

  • dt – time step size (need to be sufficiently small)

  • u0 – initial value

  • v0 – initial velocity

  • t0 – initial time

getTimeStepSize()
update(t)

returns the solution for the next time marker t which needs to greater than the time marker from the previous call.

class esys.downunder.WennerSurvey(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)

WennerSurvey forward calculation

__init__(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

returns a list containing 3 lists one for each the primary, secondary and total potential.

esys.downunder.xrange

alias of range

Functions

esys.downunder.CartesianCoordinateTransformation(domain, reference=<esys.downunder.coordinates.CartesianReferenceSystem object>)
esys.downunder.GRS80ReferenceSystem()

returns the GeodeticReferenceSystem for the GRS80 Ellipsoid eg. used by Geocentric Datum of Australia GDA94

esys.downunder.SphericalReferenceSystem(R=6378137.0)

returns the GeodeticReferenceSystem of a sphere :param R: sphere radius :type R: positive double

esys.downunder.WGS84ReferenceSystem()

returns the GeodeticReferenceSystem for the WGS84 Ellipsoid

esys.downunder.createAbsorptionLayerFunction(x, absorption_zone=300.0, absorption_cut=0.01, top_absorption=False)

Creates a distribution which is one in the interior of the domain of x and is falling down to the value ‘absorption_cut’ over a margin of thickness ‘absorption_zone’ toward each boundary except the top of the domain.

Parameters

  • x (escript.Data) – location of points in the domain

  • absorption_zone – thickness of the absorption zone

  • absorption_cut – value of decay function on domain boundary

Returns

function on ‘x’ which is one in the iterior and decays to almost zero over a margin toward the boundary.

esys.downunder.makeTransformation(domain, coordinates=None)

returns a SpatialCoordinateTransformation for the given domain

Parameters
Returns

the spatial coordinate system for the given domain of the specified reference system coordinates. If coordinates is already spatial coordinate system based on the riven domain coordinates is returned. Otherwise an appropriate spatial coordinate system is created.

Return type

SpatialCoordinateTransformation

esys.downunder.saveSilo(filename, domain=None, write_meshdata=False, time=0.0, cycle=0, **data)

Writes Data objects and their mesh to a file using the SILO file format.

Example:

temp=Scalar(..)
v=Vector(..)
saveSilo("solution.silo", temperature=temp, velocity=v)

temp and v are written to “solution.silo” where temp is named “temperature” and v is named “velocity”.

Parameters

  • filename (str) – name of the output file (‘.silo’ is added if required)

  • domain (escript.Domain) – domain of the Data objects. If not specified, the domain of the given Data objects is used.

  • write_meshdata (bool) – whether to save mesh-related data such as element identifiers, ownership etc. This is mainly useful for debugging.

  • time (float) – the timestamp to save within the file

  • cycle (int) – the cycle (or timestep) of the data

  • <name> – writes the assigned value to the Silo file using <name> as identifier

Note

All data objects have to be defined on the same domain but they may be defined on separate FunctionSpace s.

Others

  • pi