# Source code for pylops.basicoperators.CausalIntegration

import numpy as np

from pylops import LinearOperator

[docs]class CausalIntegration(LinearOperator):
r"""Causal integration.

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

Parameters
----------
N : :obj:int
Number of samples in model.
dims : :obj:list, optional
Number of samples for each dimension
(None if only one dimension is available)
dir : :obj:int, optional
Direction along which smoothing is applied.
sampling : :obj:float, optional
Sampling step dx.
halfcurrent : :obj: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 : :obj:str, optional
Type of elements in input array.
kind : :obj:str, optional
Integration kind (full, half, or trapezoidal).
removefirst : :obj:bool, optional
Remove first sample (True) or not (False).

Attributes
----------
shape : :obj:tuple
Operator shape
explicit : :obj:bool
Operator contains a matrix that can be solved explicitly (True)
or not (False)

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:

.. math::
y(t) = \int\limits_{-\infty}^t x(\tau) \,\mathrm{d}\tau

which can be discretised as :

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

or

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

or

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

where :math:\Delta t is the sampling interval, and assuming the signal is zero
before sample :math:j=0. In our implementation, the
choice to add :math:x[i] or :math:0.5x[i] is made by selecting kind=full
or kind=half, respectively. The choice to add :math:0.5x[i] and
:math: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 :math:x(\tau) = t^2 the
resulting indefinite integration is:

.. math::
y(t) = \int \tau^2 \,\mathrm{d}\tau = \frac{t^3}{3} + C

However, if we apply a first derivative to :math:y always obtain:

.. math::
x(t) = \frac{\mathrm{d}y}{\mathrm{d}t} = t^2

no matter the choice of :math:C.

"""

def __init__(
self,
N,
dims=None,
dir=-1,
sampling=1,
halfcurrent=True,
dtype="float64",
kind="full",
removefirst=False,
):
self.N = N
self.dir = dir
self.sampling = sampling
self.kind = kind
if kind == "full" and halfcurrent:  # ensure backcompatibility
self.kind = "half"
self.removefirst = removefirst
# define samples to remove from output
rf = 0
if removefirst:
rf = 1 if dims is None else self.N // dims[self.dir]
if dims is None:
self.dims = [self.N, 1]
self.dimsd = [self.N - rf, 1]
self.reshape = False
else:
if np.prod(dims) != self.N:
raise ValueError("product of dims must equal N!")
else:
self.dims = dims
self.dimsd = list(dims)
if self.removefirst:
self.dimsd[self.dir] -= 1
self.reshape = True
self.shape = (self.N - rf, self.N)
self.dtype = np.dtype(dtype)
self.explicit = False

def _matvec(self, x):
if self.reshape:
x = np.reshape(x, self.dims)
if self.dir != -1:
x = np.swapaxes(x, self.dir, -1)
y = self.sampling * np.cumsum(x, axis=-1)
if self.kind in ("half", "trapezoidal"):
y -= self.sampling * x / 2.0
if self.kind == "trapezoidal":
y[..., 1:] -= self.sampling * x[..., 0:1] / 2.0
if self.removefirst:
y = y[..., 1:]
if self.dir != -1:
y = np.swapaxes(y, -1, self.dir)
return y.ravel()

def _rmatvec(self, x):
if self.reshape:
x = np.reshape(x, self.dimsd)
if self.removefirst:
x = np.insert(x, 0, 0, axis=self.dir)
if self.dir != -1:
x = np.swapaxes(x, self.dir, -1)
xflip = np.flip(x, axis=-1)
if self.kind == "half":
y = self.sampling * (np.cumsum(xflip, axis=-1) - xflip / 2.0)
elif self.kind == "trapezoidal":
y = self.sampling * (np.cumsum(xflip, axis=-1) - xflip / 2.0)
y[..., -1] = self.sampling * np.sum(xflip, axis=-1) / 2.0
else:
y = self.sampling * np.cumsum(xflip, axis=-1)
y = np.flip(y, axis=-1)
if self.dir != -1:
y = np.swapaxes(y, -1, self.dir)
return y.ravel()