"""Vening Meinesz kernel."""
import astropy.units as u
import numpy as np
from pygeoid.integrals.core import SphericalKernel
__all__ = ["VeningMeineszKernel"]
[docs]
class VeningMeineszKernel(SphericalKernel):
r"""Vening Meinesz kernel class."""
_name = "Vening Meinesz"
[docs]
@u.quantity_input
def kernel(self, spherical_distance: u.deg):
r"""Evaluate Vening Meinesz kernel.
Parameters
----------
spherical_distance : float or array_like of floats
Spherical distance, in radians.
degrees : bool, optional
If True, the spherical distance is given in degrees,
otherwise radians.
Notes
-----
The derivative is the Vening-Meinesz and it depends
on the spherical distance :math:`\psi` by [1]_:
.. math ::
V\left(\psi\right) =
\dfrac{d S\left(\psi\right)}{d\psi} = - \dfrac{\cos{(\psi /
2)}}{2\sin^2{(\psi / 2)}} + 8\sin{\psi} -
6\cos{(\psi / 2)} - 3\dfrac{1 - \sin{(\psi / 2)}}{\sin{\psi}} +
3\sin{\psi}\ln{\left[\sin{(\psi/2)} + \sin^2{(\psi/2)}\right]},
which is the derivative of Stokes's kernel with respect to
:math:`\psi`.
References
----------
.. [1] Heiskanen WA, Moritz H (1967) Physical geodesy.
Freeman, San Francisco
"""
psi = self._check_spherical_distance(spherical_distance=spherical_distance)
cpsi2 = np.cos(psi / 2)
spsi2 = np.sin(psi / 2)
spsi = np.sin(psi)
return (
-0.5 * cpsi2 / spsi2**2
+ 8 * spsi
- 6 * cpsi2
- 3 * (1 - spsi2) / spsi
+ 3 * spsi * np.log(spsi2 + spsi2**2)
)