def test_potential():
ell = LevelEllipsoid('GRS80')
n_test = 100 # * 2
r_ = np.geomspace(ell.b, 1e8, num=n_test)
r_ = np.append(-r_[::-1], r_)
x, y, z = np.meshgrid(r_, r_, r_, indexing='ij')
sphlat, _, radius = transform.cartesian_to_spherical(x, y, z)
cond = radius < ell.polar_equation(sphlat)
x_ = np.ma.masked_where(cond, x).compressed()
y_ = np.ma.masked_where(cond, y).compressed()
z_ = np.ma.masked_where(cond, z).compressed()
sphlat, _, radius = transform.cartesian_to_spherical(x_, y_, z_)
rlat, _, u = transform.cartesian_to_ellipsoidal(x_, y_, z_, ell=ell)
gravity_potential_ell = ell.gravity_potential(rlat, u)
gravity_potential_sph = ell.gravity_potential_sph(sphlat, radius)
np.testing.assert_almost_equal(
gravity_potential_sph,
gravity_potential_ell, decimal=4)
gravitational_potential_ell = ell.gravitational_potential(rlat, u)
gravitational_potential_sph = ell.gravitational_potential_sph(
sphlat, radius)
np.testing.assert_almost_equal(
gravitational_potential_sph,
gravitational_potential_ell, decimal=4)
def gravity_disturbance(self,
lat: u.deg,
lon: u.deg,
r: u.m,
lmax: int = None) -> u.mGal:
"""Return gravity disturbance.
The gravity disturbance is defined as the magnitude of the gradient of
the potential at a given point minus the magnitude of the gradient of
the normal potential at the same point (eqs. 87 and 121 − 124 of STR09/02).
Parameters
----------
lat : ~astropy.units.Quantity
Spherical latitude.
lon : ~astropy.units.Quantity
Spherical longitude.
r : ~astropy.units.Quantity
Radial distance.
lmax : int, optional
Maximum degree of the coefficients. Default is `None` (use all
the coefficients).
Returns
-------
~astropy.units.Quantity
Gravity disturbance.
"""
rlat, _, u_ax = _transform.cartesian_to_ellipsoidal(
*_transform.spherical_to_cartesian(lat, lon, r), self._ell)
g = self._gravity.gradient(lat, lon, r, lmax)[-1]
gamma = self._ell.normal_gravity(rlat, u_ax)
return g - gamma
def height_anomaly_ell(self,
lat: u.deg,
lon: u.deg,
r: u.m,
ref_pot: u.m**2 / u.s**2 = None,
lmax=None) -> u.m:
"""Return height anomaly above the ellispoid.
The height anomaly can be generalised to a 3-d function, (sometimes
called "generalised pseudo-height-anomaly"). Here it can be calculated
on (h=0) or above (h>0) the ellipsoid, approximated by Bruns’ formula
(eqs. 78 and 118 of STR09/02)
Parameters
----------
lat : ~astropy.units.Quantity
Spherical latitude.
lon : ~astropy.units.Quantity
Spherical longitude.
r : ~astropy.units.Quantity
Radial distance.
ref_pot : ~astropy.units.Quantity
Reference potential value W0 for the zero degree term. Defaut is
`None` (zero degree term is not considered).
lmax : int, optional
Maximum degree of the coefficients. Default is `None` (use all
the coefficients).
Returns
-------
~astropy.units.Quantity
Anomaly height.
"""
rlat, _, u_ax = _transform.cartesian_to_ellipsoidal(
*_transform.spherical_to_cartesian(lat, lon, r), self._ell)
T = self.anomalous_potential.potential(lat, lon, r, lmax=lmax)
gamma = self._ell.normal_gravity(rlat, u_ax)
zeta = _np.squeeze(T / gamma)
if ref_pot is not None:
zeta -= (ref_pot - self._ell.surface_potential) / gamma
return zeta
def gravity_disturbance(self, lat, lon, r, lmax=None, degrees=True):
"""Return gravity disturbance.
The gravity disturbance is defined as the magnitude of the gradient of
the potential at a given point minus the magnitude of the gradient of
the normal potential at the same point (eqs. 87 and 121 − 124 of STR09/02).
Parameters
----------
lat : float
Latitude, in degrees
lon : float
Longitude, in degrees
r : float
Radial distance, im meters
lmax : int, optional
Maximum degree of the coefficients. Default is `None` (use all
the coefficients).
degrees : bool, optional
If True, the input `lat` and `lon` are given in degrees,
otherwise radians.
Returns
-------
float
Gravity disturbance, in m/s**2
"""
if degrees:
lat = _np.radians(lat)
lon = _np.radians(lon)
rlat, _, u = _transform.cartesian_to_ellipsoidal(
*_transform.spherical_to_cartesian(lat, lon, r, degrees=False),
self._ell,
degrees=False)
g = self._gravity.gradient(lat, lon, r, lmax, degrees=False)[-1]
gamma = self._ell.normal_gravity(rlat, u, degrees=False)
return g - gamma
def ellipsoidal(self, ell):
"""Return ellipsoidal-harmonic coordinates.
Parameters
----------
ell : instance of the `pygeoid.coordinates.ellipsoid.Ellipsoid`
Reference ellipsoid to which ellipsoidal coordinates are referenced to.
Returns
-------
rlat : ~astropy.units.Quantity
Reduced latitude.
lon : ~astropy.units.Quantity
Longitude.
u_ax : ~astropy.units.Quantity
Polar axis of the ellipsoid passing through the given point.
"""
rlat, lon, u_ax = transform.cartesian_to_ellipsoidal(self._x,
self._y,
self._z,
ell=ell)
return rlat, lon, u_ax
def height_anomaly_ell(self,
lat,
lon,
r,
ref_pot=None,
lmax=None,
degrees=True):
"""Return height anomaly above the ellispoid.
The height anomaly can be generalised to a 3-d function, (sometimes
called "generalised pseudo-height-anomaly"). Here it can be calculated
on (h=0) or above (h>0) the ellipsoid, approximated by Bruns’ formula
(eqs. 78 and 118 of STR09/02)
Parameters
----------
lat : float
Latitude, in degrees
lon : float
Longitude, in degrees
r : float
Radial distance, im meters
ref_pot : float, in m**2 / s**2
Reference potential value W0 for the zero degree term. Defaut is
`None` (zero degree term is not considered).
lmax : int, optional
Maximum degree of the coefficients. Default is `None` (use all
the coefficients).
degrees : bool, optional
If True, the input `lat` and `lon` are given in degrees,
otherwise radians.
Returns
-------
float
Anomaly height, in meters
"""
if degrees:
lat = _np.radians(lat)
lon = _np.radians(lon)
rlat, _, u = _transform.cartesian_to_ellipsoidal(
*_transform.spherical_to_cartesian(lat, lon, r, degrees=False),
self._ell,
degrees=False)
T = self.anomalous_potential.potential(lat,
lon,
r,
lmax=lmax,
degrees=False)
gamma = self._ell.normal_gravity(rlat, u, degrees=False)
zeta = _np.squeeze(T / gamma)
if ref_pot is not None:
zeta -= (ref_pot - self._ell.surface_potential) / gamma
return zeta