# Copyright (c) 2022 The Choclo Developers.
# Distributed under the terms of the BSD 3-Clause License.
# SPDX-License-Identifier: BSD-3-Clause
#
# This code is part of the Fatiando a Terra project (https://www.fatiando.org)
#
"""
Kernel functions for point sources
"""
from numba import jit
[docs]
@jit(nopython=True)
def kernel_pot(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
The inverse of the distance between the two points
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_V(\mathbf{p}, \mathbf{q}) =
\frac{
1
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 1 / distance
[docs]
@jit(nopython=True)
def kernel_e(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Easting component of the gradient of the inverse of the distance
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_x(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial x}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
- \frac{
x_p - x_q
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return -(easting_p - easting_q) / distance**3
[docs]
@jit(nopython=True)
def kernel_n(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Northing component of the gradient of the inverse of the distance
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_y(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial y}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
- \frac{
y_p - y_q
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return -(northing_p - northing_q) / distance**3
[docs]
@jit(nopython=True)
def kernel_u(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Upward component of the gradient of the inverse of the distance
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_z(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial z}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
- \frac{
z_p - z_q
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return -(upward_p - upward_q) / distance**3
[docs]
@jit(nopython=True)
def kernel_ee(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along easting-easting
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{xx}(\mathbf{p}, \mathbf{q}) =
\frac{\partial^2}{\partial x^2}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (x_p - x_q)^2
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
- \frac{
1
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^3
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (easting_p - easting_q) ** 2 / distance**5 - 1 / distance**3
[docs]
@jit(nopython=True)
def kernel_nn(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along northing-northing
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{yy}(\mathbf{p}, \mathbf{q}) =
\frac{\partial^2}{\partial y^2}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (y_p - y_q)^2
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
- \frac{
1
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^3
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (northing_p - northing_q) ** 2 / distance**5 - 1 / distance**3
[docs]
@jit(nopython=True)
def kernel_uu(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along upward-upward
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{zz}(\mathbf{p}, \mathbf{q}) =
\frac{\partial^2}{\partial z^2}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (z_p - z_q)^2
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
- \frac{
1
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^3
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (upward_p - upward_q) ** 2 / distance**5 - 1 / distance**3
[docs]
@jit(nopython=True)
def kernel_en(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along easting-northing
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{xy}(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial x \partial y}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (x_p - x_q) (y_p - y_q)
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (easting_p - easting_q) * (northing_p - northing_q) / distance**5
[docs]
@jit(nopython=True)
def kernel_eu(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along easting-upward
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{xz}(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial x \partial z}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (x_p - x_q) (z_p - z_q)
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (easting_p - easting_q) * (upward_p - upward_q) / distance**5
[docs]
@jit(nopython=True)
def kernel_nu(
easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, distance
):
r"""
Second derivative of the inverse of the distance along northing-upward
.. important ::
The coordinates of the two points must be in Cartesian coordinates and
have the same units.
Parameters
----------
easting_p, northing_p, upward_p : float
Easting, northing and upward coordinates of point :math:`\mathbf{p}`.
easting_q, northing_q, upward_q : float
Easting, northing and upward coordinates of point :math:`\mathbf{q}`.
distance : float
Euclidean distance between points :math:`\mathbf{p}` and
:math:`\mathbf{q}`.
Returns
-------
kernel : float
Value of the kernel function.
Notes
-----
Given two points :math:`\mathbf{p} = (x_p, y_p, z_p)` and :math:`\mathbf{q}
= (x_q, y_q, z_q)` defined in a Cartesian coordinate system, compute the
following kernel function:
.. math::
k_{yz}(\mathbf{p}, \mathbf{q}) =
\frac{\partial}{\partial y \partial z}
\left(
\frac{1}{\lVert \mathbf{p} - \mathbf{q} \rVert_2}
\right)
=
\frac{
3 (y_p - y_q) (z_p - z_q)
}{
\lVert \mathbf{p} - \mathbf{q} \rVert_2^5
}
where :math:`\lVert \cdot \rVert_2` refer to the :math:`L_2` norm (the
Euclidean distance between :math:`\mathbf{p}` and :math:`\mathbf{q}`).
"""
return 3 * (northing_p - northing_q) * (upward_p - upward_q) / distance**5