class pylops.CausalIntegration(N, dims=None, dir=-1, sampling=1, halfcurrent=True, dtype='float64', kind='full', removefirst=False)[source]

Causal integration.

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

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 : bool, optional

Add half of current value (True) or the entire value (False). This will be deprecated in v2.0.0, use instead kind=half to obtain the same behaviour.

dtype : str, optional

Type of elements in input array.

kind : str, optional

Integration kind (full, half, or trapezoidal).

removefirst : bool, optional

Remove first sample (True) or not (False).


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 :

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


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


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

where \(dt\) is the sampling interval. In our implementation, the choice to add \(x[i]\) or \(0.5x[i]\) is made by selecting kind=full or kind=half, respectively. The choice to add \(0.5x[i]\) and \(0.5x[0]\) instead of made by selecting the kind=trapezoidal.

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 \(c\).

shape : tuple

Operator shape

explicit : bool

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


__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