#!/usr/bin/env python
u"""
geoid_undulation.py
Written by Tyler Sutterley (04/2022)
Calculates the geoidal undulation at a given latitude and longitude using an
iterative approach described in Barthelmes (2009) and Moazezi (2012)
CALLING SEQUENCE:
geoid = geoid_undulation(lat, lon, 'WGS84', clm, slm, lmax, R, GM)
INPUT:
lat: latitude in degrees
lon: latitude in degrees
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
clm: cosine spherical harmonics for a gravity model
slm: sine spherical harmonics for a gravity model
lmax: maximum spherical harmonic degree
R: average radius used in gravity model
GM: geocentric gravitational constant used in gravity model
OPTIONS:
GAUSS: Gaussian Smoothing Radius in km (default is no filtering)
EPS: level of precision for calculating geoid height
OUTPUT:
N: geoidal undulation for a given ellipsoid in meters
PYTHON DEPENDENCIES:
numpy: Scientific Computing Tools For Python
https://numpy.org
https://numpy.org/doc/stable/user/numpy-for-matlab-users.html
PROGRAM DEPENDENCIES:
real_potential.py: real potential at lat, lon and height for gravity model
norm_potential.py: normal potential of an ellipsoid at a latitude and height
norm_gravity.py: normal gravity of an ellipsoid at a latitude and height
ref_ellipsoid.py: Computes parameters for a reference ellipsoid
gauss_weights.py: Computes Gaussian weights as a function of degree
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 11/2020: added function docstrings
Updated 07/2017: added Gaussian smoothing with option GAUSS
Written 07/2017
"""
import numpy as np
from geoid_toolkit.real_potential import real_potential
from geoid_toolkit.norm_potential import norm_potential
from geoid_toolkit.norm_gravity import norm_gravity
[docs]
def geoid_undulation(lat, lon, refell, clm, slm, lmax, R, GM, GAUSS=0, EPS=1e-8):
"""
Calculates the geoidal undulation using the iterative approach described in
:cite:t:`Barthelmes:2013fy,HofmannWellenhof:2006hy,Moazezi:2012fb`
Parameters
----------
lat: float
latitude in degrees
lon: float
longitude in degrees
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
clm: float
cosine spherical harmonics for a gravity model
slm: float
sine spherical harmonics for a gravity model
lmax: int
maximum spherical harmonic degree
R: float
average radius used in gravity model
GM: float
geocentric gravitational constant used in gravity model
GAUSS: float, default 0
Gaussian Smoothing Radius in km
EPS: float, default 1e-8
level of precision for calculating geoid height
Returns
-------
N: float
geoidal undulation for a given ellipsoid in meters
"""
# calculate the real and normal potentials for the first iteration
W,dWdr = real_potential(lat,lon,0.0,refell,clm,slm,lmax,R,GM,GAUSS=GAUSS)
U,dUdr,dUdt = norm_potential(lat, lon, 0.0, refell, lmax)
# normal gravity at latitude
gamma_h,dgamma_dh = norm_gravity(lat, 0.0, refell)
# geoid height for first iteration
N_1 = (W - U) / gamma_h
# set geoid height to the first iteration and set RMS as infinite
N = np.copy(N_1)
RMS = np.inf
while (RMS > EPS):
# calculate the real potentials for the iteration
W,dWdr=real_potential(lat,lon,N_1,refell,clm,slm,lmax,R,GM,GAUSS=GAUSS)
# add geoid height for iteration
N_1 += (W - U) / gamma_h
# calculate RMS between iterations
RMS = np.sqrt(np.sum((N - N_1)**2)/len(lat))
# set N to the previous iteration
N = np.copy(N_1)
# return the geoid height
return N