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# -*- coding: utf-8 -*- |
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# vim:fileencoding=utf-8 |
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# |
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# Copyright (c) 2017-2018 Stefan Bender |
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# |
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# This file is part of sciapy. |
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# sciapy is free software: you can redistribute it or modify it |
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# under the terms of the GNU General Public License as published by |
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# the Free Software Foundation, version 2. |
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# See accompanying LICENSE file or http://www.gnu.org/licenses/gpl-2.0.html. |
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"""IGRF geomagnetic coordinates |
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This is a python (mix) version of GMPOLE and GMCOORD from |
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http://www.ngdc.noaa.gov/geomag/geom_util/utilities_home.shtml |
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to transform geodetic to geomagnetic coordinates. |
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It uses the IGRF 2012 model and coefficients [#]_. |
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.. [#] Thébault et al. 2015, |
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International Geomagnetic Reference Field: the 12th generation. |
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Earth, Planets and Space, 67 (79) |
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http://nora.nerc.ac.uk/id/eprint/511258 |
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https://doi.org/10.1186/s40623-015-0228-9 |
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""" |
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from __future__ import absolute_import, division, print_function |
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import logging |
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from collections import namedtuple |
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from pkg_resources import resource_filename |
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import numpy as np |
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from scipy.interpolate import interp1d |
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from scipy.special import lpmn |
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__all__ = ['gmpole', 'gmag_igrf'] |
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# The WGS84 reference ellipsoid |
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Earth_ellipsoid = { |
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"a": 6378.137, # semi-major axis of the ellipsoid in km |
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"b": 6356.7523142, # semi-minor axis of the ellipsoid in km |
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"fla": 1. / 298.257223563, # flattening |
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"re": 6371.2 # Earth's radius in km |
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} |
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def _ellipsoid(ellipsoid_data=Earth_ellipsoid): |
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# extends the dictionary with the eccentricities |
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ell = namedtuple('ellip', ["a", "b", "fla", "eps", "epssq", "re"]) |
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ell.a = ellipsoid_data["a"] |
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ell.b = ellipsoid_data["b"] |
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ell.fla = ellipsoid_data["fla"] |
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ell.re = ellipsoid_data["re"] |
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# first eccentricity squared |
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ell.epssq = 1. - ell.b**2 / ell.a**2 |
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# first eccentricity |
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ell.eps = np.sqrt(ell.epssq) |
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return ell |
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def _date_to_frac_year(year, month, day): |
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# fractional year by dividing the day of year by the overall |
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# number of days in that year |
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extraday = 0 |
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if ((year % 4 == 0) and (year % 100 != 0)) or (year % 400 == 0): |
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extraday = 1 |
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month_days = [0, 31, 28 + extraday, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31] |
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doy = np.sum(month_days[:month]) + day |
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return year + (doy - 1) / (365.0 + extraday) |
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def _load_igrf_file(filename="IGRF.tab"): |
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"""Load IGRF coefficients |
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Parameters |
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---------- |
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filename: str, optional |
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The file with the IGRF coefficients. |
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Returns |
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------- |
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interpol: `scipy.interpolate.interp1d` instance |
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Interpolator instance, called with the fractional |
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year to obtain the IGRF coefficients for the epoch. |
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""" |
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igrf_tab = np.genfromtxt(filename, skip_header=3, dtype=None) |
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sv = igrf_tab[igrf_tab.dtype.names[-1]][1:].astype(np.float) |
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years = np.asarray(igrf_tab[0].tolist()[3:-1]) |
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years = np.append(years, [years[-1] + 5]) |
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coeffs = [] |
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for i in range(1, len(igrf_tab)): |
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coeff = np.asarray(igrf_tab[i].tolist()[3:-1]) |
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coeff = np.append(coeff, np.array([5]) * sv[0] + coeff[-1]) |
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coeffs.append(coeff) |
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return interp1d(years, coeffs) |
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def _geod_to_spher(phi, lon, Ellip, HeightAboveEllipsoid=0.): |
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"""Convert geodetic to spherical coordinates |
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Converts geodetic coordinates on the WGS-84 reference ellipsoid |
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to Earth-centered Earth-fixed Cartesian coordinates, |
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and then to spherical coordinates. |
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""" |
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CosLat = np.cos(np.radians(phi)) |
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SinLat = np.sin(np.radians(phi)) |
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# compute the local radius of curvature on the WGS-84 reference ellipsoid |
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rc = Ellip.a / np.sqrt(1.0 - Ellip.epssq * SinLat**2) |
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# compute ECEF Cartesian coordinates of specified point (for longitude=0) |
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xp = (rc + HeightAboveEllipsoid) * CosLat |
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zp = (rc * (1.0 - Ellip.epssq) + HeightAboveEllipsoid) * SinLat |
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# compute spherical radius and angle phi of specified point |
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rg = np.sqrt(xp**2 + zp**2) |
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# geocentric latitude |
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phig = np.degrees(np.arcsin(zp / rg)) |
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return phig, lon, rg |
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def _igrf_model(coeffs, Lmax, r, theta, phi, R_E=Earth_ellipsoid["re"]): |
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"""Evaluates the IGRF model function at the given location |
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""" |
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rho = R_E / r |
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sin_theta = np.sin(theta) |
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cos_theta = np.cos(theta) |
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Plm, dPlm = lpmn(Lmax, Lmax, cos_theta) |
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logging.debug("R_E: %s, r: %s, rho: %s", R_E, r, rho) |
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logging.debug("rho: %s, theta: %s, sin_theta: %s, cos_theta: %s", |
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rho, theta, sin_theta, cos_theta) |
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Bx, By, Bz = 0., 0., 0. # Btheta, Bphi, Br |
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idx = 0 |
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rho_l = rho |
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K_l1 = 1. |
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for l in range(1, Lmax + 1): |
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rho_l *= rho # rho^(l+1) |
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# m = 0 values, h_l^m = 0 |
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Bxl = rho_l * coeffs[idx] * dPlm[0, l] * (-sin_theta) |
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Byl = 0. |
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Bzl = -(l + 1) * rho_l * coeffs[idx] * Plm[0, l] |
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if l > 1: |
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K_l1 *= np.sqrt((l - 1) / (l + 1)) |
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idx += 1 |
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K_lm = -K_l1 |
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for m in range(1, l + 1): |
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cfi = K_lm * rho_l * (coeffs[idx] * np.cos(m * phi) |
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+ coeffs[idx + 1] * np.sin(m * phi)) |
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Bxl += cfi * dPlm[m, l] * (-sin_theta) |
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Bzl += -(l + 1) * cfi * Plm[m, l] |
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if sin_theta != 0: |
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Byl += K_lm * rho_l * m * Plm[m, l] * ( |
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- coeffs[idx] * np.sin(m * phi) + |
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coeffs[idx + 1] * np.cos(m * phi)) |
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else: |
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Byl += 0. |
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if m < l: |
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# K_lm for the next m |
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K_lm /= -np.sqrt((l + m + 1) * (l - m)) |
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idx += 2 |
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Bx += Bxl |
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By += Byl |
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Bz += Bzl |
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Bx = rho * Bx |
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By = -rho * By / sin_theta |
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Bz = rho * Bz |
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return Bx, By, Bz |
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def igrf_mag(date, lat, lon, alt, filename="IGRF.tab"): |
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"""Evaluate the local magnetic field using the IGRF model |
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Evaluates the IGRF coefficients to calculate the Earth's |
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magnetic field at the given location. |
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The results agree to within a few decimal places with |
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https://ngdc.noaa.gov/geomag-web/ |
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Parameters |
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---------- |
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date: `datetime.date` or `datetime.datetime` instance |
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The date for the evaluation. |
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lat: float |
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Geographic latitude in degrees north |
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lon: float |
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Geographic longitude in degrees east |
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alt: float |
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Altitude above ground in km. |
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filename: str, optional |
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File containing the IGRF coefficients. |
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Returns |
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------- |
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Bx: float |
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Northward component of the magnetic field, B_N. |
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By: float |
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Eastward component of the magnetic field, B_E. |
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Bz: float |
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Downward component of the magnetic field, B_D. |
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""" |
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ellip = _ellipsoid() |
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# date should be datetime.datetime or datetime.date instance, |
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# or something else that provides .year, .month, and .day attributes |
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frac_year = _date_to_frac_year(date.year, date.month, date.day) |
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glat, glon, grad = _geod_to_spher(lat, lon, ellip, alt) |
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sin_theta = np.sin(np.radians(90. - glat)) |
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cos_theta = np.cos(np.radians(90. - glat)) |
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rho = np.sqrt((ellip.a * sin_theta)**2 + (ellip.b * cos_theta)**2) |
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r = np.sqrt(alt**2 + 2 * alt * rho + |
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(ellip.a**4 * sin_theta**2 + ellip.b**4 * cos_theta**2) / rho**2) |
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cd = (alt + rho) / r |
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sd = (ellip.a**2 - ellip.b**2) / rho * cos_theta * sin_theta / r |
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logging.debug("rho: %s, r: %s, (alt + rho) / r: %s, R_E / (R_E + h): %s, R_E / r: %s", |
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rho, r, cd, ellip.re / (ellip.re + alt), ellip.re / r) |
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cos_theta, sin_theta = cos_theta*cd - sin_theta*sd, sin_theta*cd + cos_theta*sd |
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logging.debug("r: %s, spherical coordinates (radius, rho, theta, lat): %s, %s, %s, %s", |
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r, grad, rho, np.degrees(np.arccos(cos_theta)), 90. - glat) |
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# evaluate the IGRF model in spherical coordinates |
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igrf_file = resource_filename(__name__, filename) |
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igrf_coeffs = _load_igrf_file(igrf_file)(frac_year) |
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Bx, By, Bz = _igrf_model(igrf_coeffs, 13, r, np.radians(90. - glat), np.radians(glon)) |
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logging.debug("spherical geomagnetic field (Bx, By, Bz): %s, %s, %s", Bx, By, Bz) |
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logging.debug("spherical dip coordinates: lat %s, lon %s", |
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np.degrees(np.arctan2(0.5 * Bz, np.sqrt(Bx**2 + By**2))), |
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np.degrees(np.arctan2(-By, Bz))) |
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# back to geodetic coordinates |
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Bx, Bz = cd * Bx + sd * Bz, cd * Bz - sd * Bx |
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logging.debug("geodetic geomagnetic field (Bx, By, Bz): %s, %s, %s", Bx, By, Bz) |
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logging.debug("geodetic dip coordinates: lat %s, lon %s", |
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np.degrees(np.arctan2(0.5 * Bz, np.sqrt(Bx**2 + By**2))), |
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np.degrees(np.arctan2(-By, Bz))) |
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return Bx, By, Bz |
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def gmpole(date, r_e=Earth_ellipsoid["re"], filename="IGRF.tab"): |
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"""Centered dipole geomagnetic pole coordinates |
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Parameters |
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---------- |
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date: datetime.datetime or datetime.date |
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r_e: float, optional |
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Earth radius to evaluate the dipole's off-centre shift. |
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filename: str, optional |
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File containing the IGRF parameters. |
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Returns |
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------- |
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(lat_n, phi_n): tuple of floats |
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Geographic latitude and longitude of the centered dipole |
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magnetic north pole. |
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(lat_s, phi_s): tuple of floats |
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Geographic latitude and longitude of the centered dipole |
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magnetic south pole. |
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(dx, dy, dz): tuple of floats |
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Magnetic variations in Earth-centered Cartesian coordinates |
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for shifting the dipole off-center. |
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(dX, dY, dZ): tuple of floats |
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Magnetic variations in centered-dipole Cartesian coordinates |
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for shifting the dipole off-center. |
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B_0: float |
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The magnitude of the magnetic field. |
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""" |
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igrf_file = resource_filename(__name__, filename) |
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gh_func = _load_igrf_file(igrf_file) |
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frac_year = _date_to_frac_year(date.year, date.month, date.day) |
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logging.debug("fractional year: %s", frac_year) |
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g10, g11, h11, g20, g21, h21, g22, h22 = gh_func(frac_year)[:8] |
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# This function finds the location of the north magnetic pole in spherical coordinates. |
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# The equations are from Wallace H. Campbell's "Introduction to Geomagnetic Fields" |
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# and Fraser-Smith 1987, Eq. (5). |
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# For the minus signs see also |
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# Laundal and Richmond, Space Sci. Rev. (2017) 206:27--59, |
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# doi:10.1007/s11214-016-0275-y: (p. 31) |
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# "The Earth’s field is such that the dipole axis points roughly southward, |
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# so that the dipole North Pole is really in the Southern Hemisphere (SH). |
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275
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1 |
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# However convention dictates that the axis of the geomagnetic dipole is |
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276
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1 |
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# positive northward, hence the negative sign in the definition of mˆ." |
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277
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1 |
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# Note that hence Phi_N in their Eq. (14) is actually Phi_S. |
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278
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1 |
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B_0_sq = g10**2 + g11**2 + h11**2 |
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279
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theta_n = np.arccos(-g10 / np.sqrt(B_0_sq)) |
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280
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1 |
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phi_n = np.arctan2(-h11, -g11) |
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281
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1 |
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lat_n = 0.5 * np.pi - theta_n |
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282
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1 |
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logging.debug("centered dipole pole coordinates " |
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283
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1 |
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"(lat, theta, phi): %s, %s, %s", |
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284
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1 |
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np.degrees(lat_n), np.degrees(theta_n), np.degrees(phi_n)) |
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285
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1 |
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286
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# calculate dipole offset according to Fraser and Smith, 1987 |
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287
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1 |
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L_0 = 2 * g10 * g20 + np.sqrt(3) * (g11 * g21 + h11 * h21) |
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288
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1 |
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L_1 = -g11 * g20 + np.sqrt(3) * (g10 * g21 + g11 * g22 + h11 * h22) |
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289
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1 |
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L_2 = -h11 * g20 + np.sqrt(3) * (g10 * h21 - h11 * g22 + g11 * h22) |
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290
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1 |
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E = (L_0 * g10 + L_1 * g11 + L_2 * h11) / (4 * B_0_sq) |
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291
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292
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1 |
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xi = (L_0 - g10 * E) / (3 * B_0_sq) |
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293
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eta = (L_1 - g11 * E) / (3 * B_0_sq) |
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294
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1 |
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zeta = (L_2 - h11 * E) / (3 * B_0_sq) |
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295
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1 |
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dx = eta * r_e |
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296
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dy = zeta * r_e |
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297
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1 |
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dz = xi * r_e |
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298
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1 |
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299
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1 |
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# Fraser-Smith 1987, Eq. (24) |
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300
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delta = np.sqrt(dx**2 + dy**2 + dz**2) |
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301
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theta_d = np.arccos(dz / delta) |
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302
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lambda_d = 0.5 * np.pi - theta_d |
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303
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1 |
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phi_d = np.arctan2(dy, dx) |
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304
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1 |
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|
305
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1 |
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sin_lat_ed = (np.sin(lambda_d) * np.sin(lat_n) |
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306
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1 |
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+ np.cos(lambda_d) * np.cos(lat_n) * np.cos(phi_d - phi_n)) |
|
307
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lat_ed = np.arcsin(sin_lat_ed) |
|
308
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theta_ed = 0.5 * np.pi - lat_ed |
|
309
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1 |
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|
|
310
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sin_lon_ed = np.sin(theta_d) * np.sin(phi_d - phi_n) / np.sin(theta_ed) |
|
311
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lon_ed = np.pi - np.arcsin(sin_lon_ed) |
|
312
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logging.debug("eccentric dipole pole coordinates " |
|
313
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"(lat, theta, lon): %s, %s, %s", |
|
314
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1 |
|
np.degrees(theta_ed), np.degrees(lat_ed), np.degrees(lon_ed)) |
|
315
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|
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|
|
316
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dX = delta * np.sin(theta_ed) * np.cos(lon_ed) |
|
317
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|
dY = delta * np.sin(theta_ed) * np.sin(lon_ed) |
|
318
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|
|
dZ = delta * np.cos(theta_ed) |
|
319
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|
logging.debug("magnetic variations (dX, dY, dZ): %s, %s, %s", dX, dY, dZ) |
|
320
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|
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|
|
321
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|
# North pole, south pole coordinates |
|
322
|
|
|
return ((np.degrees(lat_n), np.degrees(phi_n)), |
|
323
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|
|
(-np.degrees(lat_n), np.degrees(phi_n + np.pi)), |
|
324
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|
|
(dx, dy, dz), |
|
325
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|
|
(dX, dY, dZ), |
|
326
|
|
|
np.sqrt(B_0_sq)) |
|
327
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|
|
328
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|
def gmag_igrf(date, lat, lon, alt=0., |
|
329
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|
centered_dipole=False, |
|
330
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|
|
igrf_name="IGRF.tab"): |
|
331
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|
|
"""Centered or eccentric dipole geomagnetic coordinates |
|
332
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|
|
|
333
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|
Parameters |
|
334
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|
|
---------- |
|
335
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|
|
date: datetime.datetime |
|
336
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|
|
lat: float |
|
337
|
|
|
Geographic latitude in degrees north |
|
338
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|
|
lon: float |
|
339
|
|
|
Geographic longitude in degrees east |
|
340
|
|
|
alt: float, optional |
|
341
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|
|
Altitude in km. Default: 0. |
|
342
|
|
|
centered_dipole: bool, optional |
|
343
|
|
|
Returns the centered dipole geomagnetic coordinates |
|
344
|
|
|
if set to True, returns the eccentric dipole |
|
345
|
1 |
|
geomagnetic coordinates if set to False. |
|
346
|
1 |
|
Default: False |
|
347
|
1 |
|
igrf_name: str, optional |
|
348
|
1 |
|
Default: "IGRF.tab" |
|
349
|
1 |
|
|
|
350
|
1 |
|
Returns |
|
351
|
|
|
------- |
|
352
|
1 |
|
geomag_latitude: numpy.ndarray or float |
|
353
|
1 |
|
Geomagnetic latitude in eccentric dipole coordinates, |
|
354
|
|
|
centered dipole coordinates if `centered_dipole` is True. |
|
355
|
1 |
|
geomag_longitude: numpy.ndarray or float |
|
356
|
1 |
|
Geomagnetic longitude in eccentric dipole coordinates, |
|
357
|
|
|
centered dipole coordinates if `centered_dipole` is True. |
|
358
|
1 |
|
""" |
|
359
|
1 |
|
ellip = _ellipsoid() |
|
360
|
1 |
|
glat, glon, grad = _geod_to_spher(lat, lon, ellip, alt) |
|
361
|
|
|
(lat_GMP, lon_GMP), _, _, (dX, dY, dZ), B_0 = gmpole(date, ellip.re, igrf_name) |
|
362
|
|
|
latr, lonr = np.radians(glat), np.radians(glon) |
|
363
|
1 |
|
lat_GMPr, lon_GMPr = np.radians(lat_GMP), np.radians(lon_GMP) |
|
364
|
1 |
|
sin_lat_gmag = (np.sin(latr) * np.sin(lat_GMPr) |
|
365
|
|
|
+ np.cos(latr) * np.cos(lat_GMPr) * np.cos(lonr - lon_GMPr)) |
|
366
|
|
|
lon_gmag_y = np.cos(latr) * np.sin(lonr - lon_GMPr) |
|
367
|
1 |
|
lon_gmag_x = (np.cos(latr) * np.sin(lat_GMPr) * np.cos(lonr - lon_GMPr) |
|
368
|
|
|
- np.sin(latr) * np.cos(lat_GMPr)) |
|
369
|
1 |
|
lat_gmag = np.arcsin(sin_lat_gmag) |
|
370
|
|
|
lat_gmag_geod = np.arctan2(np.tan(lat_gmag), (1. - ellip.epssq)) |
|
371
|
|
|
|
|
372
|
|
|
B_r = -2. * B_0 * sin_lat_gmag / (grad / ellip.re)**3 |
|
373
|
1 |
|
B_th = -B_0 * np.cos(lat_gmag) / (grad / ellip.re)**3 |
|
374
|
|
|
logging.debug("B_r: %s, B_th: %s, dip lat: %s", |
|
375
|
1 |
|
-B_r, B_th, np.degrees(np.arctan2(0.5 * B_r, B_th))) |
|
376
|
|
|
|
|
377
|
1 |
|
lon_gmag = np.arctan2(lon_gmag_y, lon_gmag_x) |
|
378
|
1 |
|
logging.debug("centered dipole coordinates: " |
|
379
|
1 |
|
"lat_gmag: %s, lon_gmag: %s", |
|
380
|
1 |
|
np.degrees(lat_gmag), np.degrees(lon_gmag)) |
|
381
|
1 |
|
logging.debug("lat_gmag_geod: %s, lat_GMPr: %s", |
|
382
|
1 |
|
np.degrees(lat_gmag_geod), np.degrees(lat_GMPr)) |
|
383
|
|
|
if centered_dipole: |
|
384
|
|
|
return (np.degrees(lat_gmag), np.degrees(lon_gmag)) |
|
385
|
|
|
|
|
386
|
|
|
# eccentric dipole coordinates (shifted dipole) |
|
387
|
|
|
phi_ed = np.arctan2( |
|
388
|
|
|
(grad * np.cos(lat_gmag) * np.sin(lon_gmag) - dY), |
|
389
|
|
|
(grad * np.cos(lat_gmag) * np.cos(lon_gmag) - dX) |
|
390
|
|
|
) |
|
391
|
|
|
theta_ed = np.arctan2( |
|
392
|
|
|
(grad * np.cos(lat_gmag) * np.cos(lon_gmag) - dX), |
|
393
|
|
|
(grad * np.sin(lat_gmag) - dZ) * np.cos(phi_ed) |
|
394
|
|
|
) |
|
395
|
|
|
lat_ed = 0.5 * np.pi - theta_ed |
|
396
|
|
|
lat_ed_geod = np.arctan2(np.tan(lat_ed), (1. - ellip.epssq)) |
|
397
|
|
|
logging.debug("lats ed: %s", np.degrees(lat_ed)) |
|
398
|
|
|
logging.debug("lats ed geod: %s", np.degrees(lat_ed_geod)) |
|
399
|
|
|
logging.debug("phis ed: %s", np.degrees(phi_ed)) |
|
400
|
|
|
return (np.degrees(lat_ed_geod), np.degrees(phi_ed)) |
|
401
|
|
|
|
st-bender /
sciapy
