pylops.CausalIntegration

class pylops.CausalIntegration(dims, axis=-1, sampling=1.0, kind='full', removefirst=False, dtype='float64', name='C')

Causal integration.

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

Parameters:

dimslist or int

Number of samples for each dimension

axisint, optional

Added in version 2.0.0.

Axis along which the model is integrated.

samplingfloat, optional

Sampling step dx.

kindstr, optional

Integration kind (full, half, or trapezoidal).

removefirstbool, optional

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

dtypestr, optional

Type of elements in input array.

namestr, optional

Added in version 2.0.0.

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

dimstuple

Shape of the array after the adjoint, but before flattening.

For example, x_reshaped = (Op.H * y.ravel()).reshape(Op.dims).

dimsdtuple

Shape of the array after the forward, but before flattening.

For example, y_reshaped = (Op * x.ravel()).reshape(Op.dimsd).

shapetuple

Operator shape.

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:

\[y(t) = \int\limits_{-\infty}^t x(\tau) \,\mathrm{d}\tau\]

which can be discretised as :

\[y[i] = \sum_{j=0}^i x[j] \,\Delta t\]

or

\[y[i] = \left(\sum_{j=0}^{i-1} x[j] + 0.5x[i]\right) \,\Delta t\]

or

\[y[i] = \left(\sum_{j=1}^{i-1} x[j] + 0.5x[0] + 0.5x[i]\right) \,\Delta t\]

where \(\Delta t\) is the sampling interval, and assuming the signal is zero before sample \(j=0\). 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 causal integral of a signal will depend, up to a constant, on causal start of the signal. For example if \(x(\tau) = t^2\) the resulting indefinite integration is:

\[y(t) = \int \tau^2 \,\mathrm{d}\tau = \frac{t^3}{3} + C\]

However, if we apply a first derivative to \(y\) always obtain:

\[x(t) = \frac{\mathrm{d}y}{\mathrm{d}t} = t^2\]

no matter the choice of \(C\).

Methods

__init__(dims[, axis, sampling, kind, ...])
adjoint()
apply_columns(cols)	Apply subset of columns of operator
cond([uselobpcg])	Condition number of linear operator.
conj()	Complex conjugate operator
div(y[, niter, densesolver])	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.
reset_count()	Reset counters
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.
trace([neval, method, backend])	Trace of linear operator.
transpose()

Examples using pylops.CausalIntegration

Causal Integration

Causal Integration