#!/usr/bin/env python
u"""
norm_gravity.py
Written by Tyler Sutterley (04/2022)
Calculates the normal gravity of an ellipsoid at a given latitude and height
and calculates the derivative with respect to height
CALLING SEQUENCE:
gamma_h, dgamma_dh = norm_gravity(latitude, height, 'WGS84')
INPUT:
latitude: latitude in degrees
height: height above reference ellipsoid in meters
refell: reference ellipsoid name
CLK66 = Clarke 1866
GRS67 = Geodetic Reference System 1967
GRS80 = Geodetic Reference System 1980
WGS72 = World Geodetic System 1972
WGS84 = World Geodetic System 1984
ATS77 = Quasi-earth centred ellipsoid for ATS77
NAD27 = North American Datum 1927
NAD83 = North American Datum 1983
INTER = International
KRASS = Krassovsky (USSR)
MAIRY = Modified Airy (Ireland 1965/1975)
TOPEX = TOPEX/POSEIDON ellipsoid
EGM96 = EGM 1996 gravity model
OUTPUT:
gamma_h: normal gravity for ellipsoid at height
dgamma_dh: derivative of normal gravity with respect to height
PYTHON DEPENDENCIES:
numpy: Scientific Computing Tools For Python
https://numpy.org
https://numpy.org/doc/stable/user/numpy-for-matlab-users.html
PROGRAM DEPENDENCIES:
ref_ellipsoid.py: Computes parameters for a reference ellipsoid
REFERENCE:
Hofmann-Wellenhof and Moritz, "Physical Geodesy" (2005)
http://www.springerlink.com/content/978-3-211-33544-4
Barthelmes, "Definition of Functionals of the Geopotential and Their
Calculation from Spherical Harmonic Models", STR09/02 (2009)
http://icgem.gfz-potsdam.de/str-0902-revised.pdf
Moazezi and Zomorrodian, "GGMCalc a software for calculation of the geoid
undulation and the height anomaly using the iteration method, and
classical gravity anomaly", Earth Science Informatics (2012)
https://doi.org/10.1007/s12145-012-0102-2
UPDATE HISTORY:
Updated 04/2022: updated docstrings to numpy documentation format
Updated 12/2020: updated comments and reorganized functions
Updated 11/2020: added function docstrings
Updated 07/2017: added header text. higher order expansion of normal gravity
Written 08/2013
"""
import numpy as np
from geoid_toolkit.ref_ellipsoid import ref_ellipsoid
[docs]
def norm_gravity(lat, h, refell):
"""
Calculates the normal gravity of an ellipsoid and calculates the derivative
with respect to height following :cite:t:`HofmannWellenhof:2006hy`
Parameters
----------
lat: float
latitude in degrees
h: float
ellipsoidal height in meters
refell: str
Reference ellipsoid name
- ``'CLK66'``: Clarke 1866
- ``'GRS67'``: Geodetic Reference System 1967
- ``'GRS80'``: Geodetic Reference System 1980
- ``'HGH80'``: Hughes 1980 Ellipsoid
- ``'WGS72'``: World Geodetic System 1972
- ``'WGS84'``: World Geodetic System 1984
- ``'ATS77'``: Quasi-earth centred ellipsoid for ATS77
- ``'NAD27'``: North American Datum 1927
- ``'NAD83'``: North American Datum 1983
- ``'INTER'``: International
- ``'KRASS'``: Krassovsky (USSR)
- ``'MAIRY'``: Modified Airy (Ireland 1965/1975)
- ``'TOPEX'``: TOPEX/POSEIDON ellipsoid
- ``'EGM96'``: EGM 1996 gravity model
Returns
-------
gamma_h: float
normal gravity for ellipsoid at height
dgamma_dh: float
derivative of normal gravity with respect to height
"""
# convert latitude from degrees to radians
phi = np.pi*lat/180.0
# get ellipsoid parameters for refell
ellip = ref_ellipsoid(refell)
a = ellip['a']
b = ellip['b']
# eccentricity
ecc2 = ellip['ecc2']
GM = ellip['GM']
# m parameter [omega^2*a^2*b/(GM)]
m = ellip['mp']
# flattening components
f = ellip['f']
f_2 = -f + (5.0/2.0)*m + (1.0/2.0)*f**2.0 - (26.0/7.0)*f*m + (15.0/4.0)*m**2.0
f_4 = -(1.0/2.0)*f**2.0 + (5.0/2.0)*f*m
# Normal gravity at the equator.
# p. 79, Eqn.(2-186)
gamma_a = (GM/(a * b)) * (1.0 - (3.0 / 2.0)*m - (3.0 / 14.0)*ecc2**2.0*m)
# Normal gravity
# p. 80, Eqn.(2-199)
gamma_0 = gamma_a * (1.0 + f_2*np.sin(phi)**2.0 + f_4*np.sin(phi)**4.0)
# Normal gravity at height
# p. 82, Eqn.(2-215)
p_1 = (1.0 + f + m - 2.0*f*np.sin(phi)**2.0)
gamma_h = gamma_0 * (1.0 - (2.0/a)*p_1*h + (3.0/(a**2.0))*h**2.0)
# approximate derivative of normal gravity with respect to height
dgamma_dh = ((-2.0 * gamma_0) / a) * p_1
# return the normal gravity and the derivative
return (gamma_h, dgamma_dh)