pylops.signalprocessing.Radon3D¶

pylops.signalprocessing.Radon3D(taxis, hyaxis, hxaxis, pyaxis, pxaxis, kind='linear', centeredh=True, interp=True, onthefly=False, engine='numpy', dtype='float64', name='R')[source]¶

Three dimensional Radon transform.

Apply three dimensional Radon forward (and adjoint) transform to a 3-dimensional array of size \([n_{p_y} \times n_{p_x} \times n_t]\) (and \([n_y \times n_x \times n_t]\)).

In forward mode this entails to spreading the model vector along parametric curves (lines, parabolas, or hyperbolas depending on the choice of kind), while stacking values in the data vector along the same parametric curves is performed in adjoint mode.

Parameters:

taxisnumpy.ndarray: Time axis
hyaxisnumpy.ndarray: Slow spatial axis
hxaxisnumpy.ndarray: False spatial axis
pyaxisnumpy.ndarray: Axis of scanning variable \(p_y\) of parametric curve
pxaxisnumpy.ndarray: Axis of scanning variable \(p_x\) of parametric curve
kindstr, optional: Curve to be used for stacking/spreading (linear, parabolic, and hyperbolic are currently supported)
centeredhbool, optional: Assume centered spatial axis (True) or not (False). If True the original haxis is ignored and a new axis is created.
interpbool, optional: Apply linear interpolation (True) or nearest interpolation (False) during stacking/spreading along parametric curve
ontheflybool, optional: Compute stacking parametric curves on-the-fly as part of forward and adjoint modelling (True) or at initialization and store them in look-up table (False). Using a look-up table is computationally more efficient but increases the memory burden
enginestr, optional: Engine used for computation (numpy or numba)
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)

Returns:

r3oppylops.LinearOperator: Radon operator

Raises:

NotImplementedError: If kind is not linear, parabolic, or hyperbolic
ValueError: If engine is neither numpy nor numba

See also

pylops.signalprocessing.Radon2D: Two dimensional Radon transform
pylops.Spread: Spread operator

Notes

The Radon3D operator applies the following linear transform in adjoint mode to the data after reshaping it into a 3-dimensional array of size \([n_y \times n_x \times n_t]\) in adjoint mode:

\[m(p_y, p_x, t_0) = \int{d(y, x, t = f(p_y, p_x, y, x, t))} \,\mathrm{d}x \,\mathrm{d}y\]

where \(f(p_y, p_x, y, x, t) = t_0 + p_y y + p_x x\) in linear mode, \(f(p_y, p_x, y, x, t) = t_0 + p_y y^2 + p_x x^2\) in parabolic mode, and \(f(p_y, p_x, y, x, t) = \sqrt{t_0^2 + y^2 / p_y^2 + x^2 / p_x^2}\) in hyperbolic mode. Note that internally the \(p_x\) and \(p_y\) axes will be normalized by the ratio of the spatial and time axes and used alongside unitless axes. Whilst this makes the linear mode fully unitless, users are required to apply additional scalings to the \(p_x\) axis for other relationships

\(p_x\) should be pre-multipled by \(d_x / d_y\) for the parabolic relationship;
\(p_x\) should be pre-multipled by \((d_t/d_x)^2 / (d_t/d_y)^2\) for the hyperbolic relationship.

As the adjoint operator can be interpreted as a repeated summation of sets of elements of the model vector along chosen parametric curves, the forward is implemented as spreading of values in the data vector along the same parametric curves. This operator is actually a thin wrapper around the pylops.Spread operator.