pylops.CausalIntegration

class pylops.CausalIntegration(N, dims=None, dir=-1, sampling=1, halfcurrent=True, dtype='float64')

Causal integration.

Apply causal integration to a multi-dimensional array along dir axis.

Parameters:

N : int

Number of samples in model.

dims : list, optional

Number of samples for each dimension (None if only one dimension is available)

dir : int, optional

Direction along which smoothing is applied.

sampling : float, optional

Sampling step dx.

halfcurrent : float, optional

Add half of current value (True) or the entire value (False)

dtype : str, optional

Type of elements in input array.

Notes

The CausalIntegration operator applies a causal integration to any chosen direction of a multi-dimensional array.

For simplicity, given a one dimensional array, the causal integration is:

\[y(t) = \int x(t) dt\]

which can be discretised as :

\[y[i] = \sum_{j=0}^i x[j] dt\]

or

\[y[i] = (\sum_{j=0}^{i-1} x[j] + 0.5x[i]) dt\]

where \(dt\) is the sampling interval. In our implementation, the choice to add \(x[i]\) or just \(0.5x[i]\) is made by selecting the halfcurrent parameter.

Note that the integral of a signal has no unique solution, as any constant \(c\) can be added to \(y\), for example if \(x(t)=t^2\) the resulting integration is:

\[y(t) = \int t^2 dt = \frac{t^3}{3} + c\]

If we apply a first derivative to \(y\) we in fact obtain:

\[x(t) = \frac{dy}{dt} = t^2\]

no matter the choice of \(c\).

Attributes:

shape : tuple

Operator shape

explicit : bool

Operator contains a matrix that can be solved explicitly (True) or not (False)

Methods

__init__(N[, dims, dir, sampling, …])	Initialize this LinearOperator.
adjoint()	Hermitian adjoint.
apply_columns(cols)	Apply subset of columns of operator
cond([uselobpcg])	Condition number of linear operator.
conj()	Complex conjugate operator
div(y[, niter])	Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot(x)	Matrix-matrix or matrix-vector multiplication.
eigs([neigs, symmetric, niter, uselobpcg])	Most significant eigenvalues of linear operator.
matmat(X)	Matrix-matrix multiplication.
matvec(x)	Matrix-vector multiplication.
rmatmat(X)	Matrix-matrix multiplication.
rmatvec(x)	Adjoint matrix-vector multiplication.
todense([backend])	Return dense matrix.
toimag([forw, adj])	Imag operator
toreal([forw, adj])	Real operator
tosparse()	Return sparse matrix.
transpose()	Transpose this linear operator.

Examples using pylops.CausalIntegration

Causal Integration