Source code for pyrocko.orthodrome

# http://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------
from __future__ import division, absolute_import
from builtins import object

import math
import numpy as num

from .moment_tensor import euler_to_matrix
from .config import config
from .plot.beachball import spoly_cut

from matplotlib.path import Path

d2r = math.pi/180.
r2d = 1./d2r
earth_oblateness = 1./298.257223563
earthradius_equator = 6378.14 * 1000.
earthradius = config().earthradius
d2m = earthradius_equator*math.pi/180.
m2d = 1./d2m

_testpath = Path([(0, 0), (0, 1), (1, 1), (1, 0), (0, 0)], closed=True)

if hasattr(_testpath, 'contains_points') and num.all(
        _testpath.contains_points([(0.5, 0.5), (1.5, 0.5)]) == [True, False]):

    def path_contains_points(verts, points):
        p = Path(verts, closed=True)
        return p.contains_points(points).astype(num.bool)

else:
    # work around missing contains_points and bug in matplotlib ~ v1.2.0

[docs] def path_contains_points(verts, points): p = Path(verts, closed=True) result = num.zeros(points.shape[0], dtype=num.bool) for i in range(result.size): result[i] = p.contains_point(points[i, :]) return result
try: cbrt = num.cbrt except AttributeError: def cbrt(x): return x**(1./3.)
[docs]def float_array_broadcast(*args): return num.broadcast_arrays(*[ num.asarray(x, dtype=num.float) for x in args])
[docs]class Loc(object): '''Simple location representation :attrib lat: Latitude degree :attrib lon: Longitude degree ''' def __init__(self, lat, lon): self.lat = lat self.lon = lon
[docs]def clip(x, mi, ma): ''' Clipping data array ``x`` :param x: Continunous data to be clipped :param mi: Clip minimum :param ma: Clip maximum :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float :return: Clipped data :rtype: :py:class:`numpy.ndarray` ''' return num.minimum(num.maximum(mi, x), ma)
[docs]def wrap(x, mi, ma): '''Wrapping continuous data to fundamental phase values. .. math:: x_{\\mathrm{wrapped}} = x_{\\mathrm{cont},i} - \ \\frac{ x_{\\mathrm{cont},i} - r_{\\mathrm{min}} }\ { r_{\\mathrm{max}} - r_{\\mathrm{min}}} \ \cdot ( r_{\\mathrm{max}} - r_{\\mathrm{min}}),\ \quad x_{\\mathrm{wrapped}}\; \in \;[ r_{\\mathrm{min}},\, r_{\\mathrm{max}}]. :param x: Continunous data to be wrapped :param mi: Minimum value of wrapped data :param ma: Maximum value of wrapped data :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float :return: Wrapped data :rtype: :py:class:`numpy.ndarray` ''' return x - num.floor((x-mi)/(ma-mi)) * (ma-mi)
def _latlon_pair(args): if len(args) == 2: a, b = args return a.lat, a.lon, b.lat, b.lon elif len(args) == 4: return args
[docs]def cosdelta(*args): '''Cosine of the angular distance between two points ``a`` and ``b`` on a sphere. This function (find implementation below) returns the cosine of the distance angle 'delta' between two points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of 1.0 is enforced. .. math:: A_{\\mathrm{lat'}} = \\frac{ \pi}{180} \cdot A_{lat}, \quad \ A_{\\mathrm{lon'}} = \\frac{ \pi}{180} \cdot A_{lon}, \quad \ B_{\\mathrm{lat'}} = \\frac{ \pi}{180} \cdot B_{lat}, \quad \ B_{\\mathrm{lon'}} = \\frac{ \pi}{180} \cdot B_{lon}\\\\[0.5cm] \\cos(\Delta) = \\min( 1.0, \quad \\sin( A_{\\mathrm{lat'}}) \ \\sin( B_{\\mathrm{lat'}} ) + \ \\cos(A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \ \\cos( B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} ) :param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc` :return: cosdelta :rtype: float ''' alat, alon, blat, blon = _latlon_pair(args) return min( 1.0, math.sin(alat*d2r) * math.sin(blat*d2r) + math.cos(alat*d2r) * math.cos(blat*d2r) * math.cos(d2r*(blon-alon)))
[docs]def cosdelta_numpy(a_lats, a_lons, b_lats, b_lons): '''Cosine of the angular distance between two points ``a`` and ``b`` on a sphere. This function returns the cosines of the distance angles *delta* between two points ``a`` and ``b`` given as :py:class:`numpy.ndarray`. The coordinates are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of ``1.0`` is enforced. Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.cosdelta` above. :param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray` :type a_lons: :py:class:`numpy.ndarray` :type b_lats: :py:class:`numpy.ndarray` :type b_lons: :py:class:`numpy.ndarray` :return: cosdelta :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` ''' return num.minimum( 1.0, num.sin(a_lats*d2r) * num.sin(b_lats*d2r) + num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.cos(d2r*(b_lons-a_lons)))
[docs]def azimuth(*args): '''Azimuth calculation This function (find implementation below) returns azimuth ... between points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees. .. math:: A_{\\mathrm{lat'}} = \\frac{ \pi}{180} \cdot A_{lat}, \quad \ A_{\\mathrm{lon'}} = \\frac{ \pi}{180} \cdot A_{lon}, \quad \ B_{\\mathrm{lat'}} = \\frac{ \pi}{180} \cdot B_{lat}, \quad \ B_{\\mathrm{lon'}} = \\frac{ \pi}{180} \cdot B_{lon}\\\\ \\varphi_{\\mathrm{azi},AB} = \\frac{180}{\pi} \\arctan \left[ \\frac{\ \\cos( A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \ \\sin(B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} )} \ {\\sin ( B_{\\mathrm{lat'}} ) - \\sin( A_{\\mathrm{lat'}} \ cosdelta) } \\right] :param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc` :return: Azimuth in degree ''' alat, alon, blat, blon = _latlon_pair(args) return r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cosdelta( alat, alon, blat, blon))
[docs]def azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta=None): '''Calculation of the azimuth (*track angle*) from a location A towards B. This function returns azimuths (*track angles*) from locations A towards B given in :py:class:`numpy.ndarray`. Coordinates are expected to be given in geographical coordinates and in degrees. Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.azimuth`. :param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Azimuths in degrees :rtype: :py:class:`numpy.ndarray`, ``(N)`` ''' if _cosdelta is None: _cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons) return r2d*num.arctan2( num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.sin(d2r*(b_lons-a_lons)), num.sin(d2r*b_lats) - num.sin(d2r*a_lats) * _cosdelta)
[docs]def azibazi(*args, **kwargs): alat, alon, blat, blon = _latlon_pair(args) if alat == blat and alon == blon: return 0., 180. implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.azibazi(alat, alon, blat, blon) cd = cosdelta(alat, alon, blat, blon) azi = r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cd) bazi = r2d*math.atan2( math.cos(blat*d2r) * math.cos(alat*d2r) * math.sin(d2r*(alon-blon)), math.sin(d2r*alat) - math.sin(d2r*blat) * cd) return azi, bazi
[docs]def azibazi_numpy(a_lats, a_lons, b_lats, b_lons, implementation='c'): a_lats, a_lons, b_lats, b_lons = float_array_broadcast( a_lats, a_lons, b_lats, b_lons) assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.azibazi_numpy(a_lats, a_lons, b_lats, b_lons) _cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons) azis = azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta) bazis = azimuth_numpy(b_lats, b_lons, a_lats, a_lons, _cosdelta) eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_eq = num.where(eq)[0] azis[ii_eq] = 0.0 bazis[ii_eq] = 180.0 return azis, bazis
[docs]def azidist_numpy(*args): '''Calculation of the azimuth (*track angle*) and the distance from locations A towards B on a sphere. The assisting functions used are :py:func:`pyrocko.orthodrome.cosdelta` and :py:func:`pyrocko.orthodrome.azimuth` :param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Azimuths in degrees, distances in degrees :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' _cosdelta = cosdelta_numpy(*args) _azimuths = azimuth_numpy(_cosdelta=_cosdelta, *args) return _azimuths, r2d*num.arccos(_cosdelta)
[docs]def distance_accurate50m(*args, **kwargs): ''' Accurate distance calculation based on a spheroid of rotation. Function returns distance in meter between points A and B, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50\,m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`. From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on: ``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1`` .. math:: F = \\frac{\pi}{180} \ \\frac{(A_{lat} + B_{lat})}{2}, \quad \ G = \\frac{\pi}{180} \ \\frac{(A_{lat} - B_{lat})}{2}, \quad \ l = \\frac{\pi}{180} \ \\frac{(A_{lon} - B_{lon})}{2} \quad \ \\\\[0.5cm] S = \\sin^2(G) \\cdot \\cos^2(l) + \ \\cos^2(F) \\cdot \\sin^2(l), \quad \quad C = \\cos^2(G) \\cdot \\cos^2(l) + \ \\sin^2(F) \\cdot \\sin^2(l) .. math:: w = \\arctan \\left( \\sqrt{ \\frac{S}{C}} \\right) , \quad \ r = \\sqrt{\\frac{S}{C} } The spherical-earth distance D between A and B, can be given with: .. math:: D_{sphere} = 2w \cdot R_{equator} The oblateness of the Earth requires some correction with correction factors h1 and h2: .. math:: h_1 = \\frac{3r - 1}{2C}, \quad \ h_2 = \\frac{3r +1 }{2S}\\\\[0.5cm] D = D_{\\mathrm{sphere}} \cdot [ 1 + h_1 \,f_{\\mathrm{oblate}} \ \cdot \\sin^2(F) \ \\cos^2(G) - h_2\, f_{\\mathrm{oblate}} \ \cdot \\cos^2(F) \\sin^2(G)] :param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc` :return: Distance in meter :rtype: float ''' alat, alon, blat, blon = _latlon_pair(args) implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.distance_accurate50m(alat, alon, blat, blon) f = (alat + blat)*d2r / 2. g = (alat - blat)*d2r / 2. h = (alon - blon)*d2r / 2. s = math.sin(g)**2 * math.cos(h)**2 + math.cos(f)**2 * math.sin(h)**2 c = math.cos(g)**2 * math.cos(h)**2 + math.sin(f)**2 * math.sin(h)**2 w = math.atan(math.sqrt(s/c)) if w == 0.0: return 0.0 r = math.sqrt(s*c)/w d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s) return d * (1. + earth_oblateness * h1 * math.sin(f)**2 * math.cos(g)**2 - earth_oblateness * h2 * math.cos(f)**2 * math.sin(g)**2)
[docs]def distance_accurate50m_numpy( a_lats, a_lons, b_lats, b_lons, implementation='c'): ''' Accurate distance calculation based on a spheroid of rotation. Function returns distance in meter between points ``a`` and ``b``, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50\,m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`. From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on: ``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1`` .. math:: F_i = \\frac{\pi}{180} \ \\frac{(a_{lat,i} + a_{lat,i})}{2}, \quad \ G_i = \\frac{\pi}{180} \ \\frac{(a_{lat,i} - b_{lat,i})}{2}, \quad \ l_i= \\frac{\pi}{180} \ \\frac{(a_{lon,i} - b_{lon,i})}{2} \\\\[0.5cm] S_i = \\sin^2(G_i) \\cdot \\cos^2(l_i) + \ \\cos^2(F_i) \\cdot \\sin^2(l_i), \quad \quad C_i = \\cos^2(G_i) \\cdot \\cos^2(l_i) + \ \\sin^2(F_i) \\cdot \\sin^2(l_i) .. math:: w_i = \\arctan \\left( \\sqrt{\\frac{S_i}{C_i}} \\right), \quad \ r_i = \\sqrt{\\frac{S_i}{C_i} } The spherical-earth distance ``D`` between ``a`` and ``b``, can be given with: .. math:: D_{\\mathrm{sphere},i} = 2w_i \cdot R_{\\mathrm{equator}} The oblateness of the Earth requires some correction with correction factors ``h1`` and ``h2``: .. math:: h_{1.i} = \\frac{3r - 1}{2C_i}, \quad \ h_{2,i} = \\frac{3r +1 }{2S_i}\\\\[0.5cm] D_{AB,i} = D_{\\mathrm{sphere},i} \cdot [1 + h_{1,i} \,f_{\\mathrm{oblate}} \ \cdot \\sin^2(F_i) \ \\cos^2(G_i) - h_{2,i}\, f_{\\mathrm{oblate}} \ \cdot \\cos^2(F_i) \\sin^2(G_i)] :param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` :return: Distances in degrees :rtype: :py:class:`numpy.ndarray`, ``(N)`` ''' a_lats, a_lons, b_lats, b_lons = float_array_broadcast( a_lats, a_lons, b_lats, b_lons) assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.distance_accurate50m_numpy( a_lats, a_lons, b_lats, b_lons) eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_neq = num.where(num.logical_not(eq))[0] if num.all(eq): return num.zeros_like(eq, dtype=num.float) def extr(x): if isinstance(x, num.ndarray) and x.size > 1: return x[ii_neq] else: return x a_lats = extr(a_lats) a_lons = extr(a_lons) b_lats = extr(b_lats) b_lons = extr(b_lons) f = (a_lats + b_lats)*d2r / 2. g = (a_lats - b_lats)*d2r / 2. h = (a_lons - b_lons)*d2r / 2. s = num.sin(g)**2 * num.cos(h)**2 + num.cos(f)**2 * num.sin(h)**2 c = num.cos(g)**2 * num.cos(h)**2 + num.sin(f)**2 * num.sin(h)**2 w = num.arctan(num.sqrt(s/c)) r = num.sqrt(s*c)/w d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s) dists = num.zeros(eq.size, dtype=num.float) dists[ii_neq] = d * ( 1. + earth_oblateness * h1 * num.sin(f)**2 * num.cos(g)**2 - earth_oblateness * h2 * num.cos(f)**2 * num.sin(g)**2) return dists
[docs]def ne_to_latlon(lat0, lon0, north_m, east_m): '''Transform local cartesian coordinates to latitude and longitude. From east and north coordinates (``x`` and ``y`` coordinate :py:class:`numpy.ndarray`) relative to a reference differences in longitude and latitude are calculated, which are effectively changes in azimuth and distance, respectively: .. math:: \\text{distance change:}\; \Delta {\\bf{a}} &= \sqrt{{\\bf{y}}^2 + \ {\\bf{x}}^2 }/ \mathrm{R_E}, \\text{azimuth change:}\; \Delta \\bf{\gamma} &= \\arctan( \\bf{x} \ / \\bf{y}). The projection used preserves the azimuths of the input points. :param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param north_m: Northing distances from origin in meters. :param east_m: Easting distances from origin in meters. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' a = num.sqrt(north_m**2+east_m**2)/earthradius gamma = num.arctan2(east_m, north_m) return azidist_to_latlon_rad(lat0, lon0, gamma, a)
[docs]def azidist_to_latlon(lat0, lon0, azimuth_deg, distance_deg): '''(Durchreichen??). ''' return azidist_to_latlon_rad( lat0, lon0, azimuth_deg/180.*num.pi, distance_deg/180.*num.pi)
[docs]def azidist_to_latlon_rad(lat0, lon0, azimuth_rad, distance_rad): ''' Absolute latitudes and longitudes are calculated from relative changes. For numerical stability a range between of ``-1.0`` and ``1.0`` is enforced for ``c`` and ``alpha``. .. math:: \Delta {\\bf a}_i \; \\text{and} \; \Delta \gamma_i \; \ \\text{are relative distances and azimuths from lat0 and lon0 for \ \\textit{i} source points of a finite source.} .. math:: \mathrm{b} &= \\frac{\pi}{2} -\\frac{\pi}{180} \;\mathrm{lat_0}\\\\ {\\bf c}_i &=\\arccos[\; \cos(\Delta {\\bf{a}}_i) \cos(\mathrm{b}) \ + |\Delta \gamma_i| \,\sin(\Delta {\\bf a}_i) \sin(\mathrm{b})\; ] \\\\ \mathrm{lat}_i &= \\frac{180}{\pi} \ \\left(\\frac{\pi}{2} - {\\bf c}_i \\right) .. math:: \\alpha_i &= \\arcsin \\left[ \; \\frac{ \sin(\Delta {\\bf a}_i ) \ \sin(|\Delta \\gamma_i|)}{\sin({\\bf c}_i)}\; \\right] \\\\ \\alpha_i &= \\begin{cases} \\alpha_i, &\\text{if} \; \cos(\Delta {\\bf a}_i) - \ \cos(\mathrm{b}) \cos({\\bf{c}}_i) > 0, \; \ \\text{else} \\\\ \pi - \\alpha_i, & \\text{if} \; \\alpha_i > 0,\; \\text{else}\\\\ -\pi - \\alpha_i, & \\text{if} \; \\alpha_i < 0. \\end{cases} \\\\ \mathrm{lon}_i &= \mathrm{lon_0} + \ \\frac{180}{\pi} \, \\frac{\Delta \gamma_i }{|\Delta \gamma_i|} \ \cdot \\alpha_i \ \\text{, with $\\alpha_i \\in [-\pi,\pi]$} :param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param distance_rad: Distances from origin in radians. :param azimuth_rad: Azimuth from radians. :type distance_rad: :py:class:`numpy.ndarray`, ``(N)`` :type azimuth_rad: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' a = distance_rad gamma = azimuth_rad b = math.pi/2.-lat0*d2r alphasign = 1. alphasign = num.where(gamma < 0, -1., 1.) gamma = num.abs(gamma) c = num.arccos(clip( num.cos(a)*num.cos(b)+num.sin(a)*num.sin(b)*num.cos(gamma), -1., 1.)) alpha = num.arcsin(clip( num.sin(a)*num.sin(gamma)/num.sin(c), -1., 1.)) alpha = num.where( num.cos(a)-num.cos(b)*num.cos(c) < 0, num.where(alpha > 0, math.pi-alpha, -math.pi-alpha), alpha) lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.) return lat, lon
[docs]def ne_to_latlon_alternative_method(lat0, lon0, north_m, east_m): '''Transform local cartesian coordinates to latitude and longitude. Like :py:func:`pyrocko.orthodrome.ne_to_latlon`, but this method (implementation below), although it should be numerically more stable, suffers problems at points which are *across the pole* as seen from the cartesian origin. .. math:: \\text{distance change:}\; \Delta {{\\bf a}_i} &=\sqrt{{\\bf{y}}^2_i + {\\bf{x}}^2_i }/ \mathrm{R_E},\\\\ \\text{azimuth change:}\; \Delta {\\bf \gamma}_i &= \\arctan( {\\bf x}_i {\\bf y}_i). \\\\ \mathrm{b} &= \\frac{\pi}{2} -\\frac{\pi}{180} \;\mathrm{lat_0}\\\\ .. math:: {{\\bf z}_1}_i &= \\cos{\\left( \\frac{\Delta {\\bf a}_i - \ \\mathrm{b}}{2} \\right)} \ \\cos {\\left( \\frac{|\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_1}_i &= \\cos{\\left( \\frac{\Delta {\\bf a}_i + \ \\mathrm{b}}{2} \\right)}\ \\sin {\\left( \\frac{|\gamma_i|}{2} \\right) }\\\\ {{\\bf z}_2}_i &= \\sin{\\left( \\frac{\Delta {\\bf a}_i - \ \\mathrm{b}}{2} \\right)}\ \\cos {\\left( \\frac{|\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_2}_i &= \\sin{\\left( \\frac{\Delta {\\bf a}_i + \ \\mathrm{b}}{2} \\right)}\ \\sin {\\left( \\frac{|\gamma_i|}{2} \\right) }\\\\ {{\\bf t}_1}_i &= \\arctan{\\left( \\frac{{{\\bf z}_1}_i} \ {{{\\bf n}_1}_i} \\right) }\\\\ {{\\bf t}_2}_i &= \\arctan{\\left( \\frac{{{\\bf z}_2}_i} \ {{{\\bf n}_2}_i} \\right) } \\\\[0.5cm] c &= \\begin{cases} 2 \cdot \\arccos \\left( {{\\bf z}_1}_i / \\sin({{\\bf t}_1}_i) \ \\right),\; \\text{if } \ |\\sin({{\\bf t}_1}_i)| > \ |\\sin({{\\bf t}_2}_i)|,\; \\text{else} \\\\ 2 \cdot \\arcsin{\\left( {{\\bf z}_2}_i / \ \\sin({{\\bf t}_2}_i) \\right)}. \\end{cases}\\\\ .. math:: {\\bf {lat}}_i &= \\frac{180}{ \pi } \\left( \\frac{\pi}{2} \ - {\\bf {c}}_i \\right) \\\\ {\\bf {lon}}_i &= {\\bf {lon}}_0 + \\frac{180}{ \pi } \ \\frac{\gamma_i}{|\gamma_i|}, \ \\text{ with}\; \gamma \\in [-\pi,\pi] :param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param north_m: Northing distances from origin in meters. :param east_m: Easting distances from origin in meters. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float :return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' b = math.pi/2.-lat0*d2r a = num.sqrt(north_m**2+east_m**2)/earthradius gamma = num.arctan2(east_m, north_m) alphasign = 1. alphasign = num.where(gamma < 0., -1., 1.) gamma = num.abs(gamma) z1 = num.cos((a-b)/2.)*num.cos(gamma/2.) n1 = num.cos((a+b)/2.)*num.sin(gamma/2.) z2 = num.sin((a-b)/2.)*num.cos(gamma/2.) n2 = num.sin((a+b)/2.)*num.sin(gamma/2.) t1 = num.arctan2(z1, n1) t2 = num.arctan2(z2, n2) alpha = t1 + t2 sin_t1 = num.sin(t1) sin_t2 = num.sin(t2) c = num.where( num.abs(sin_t1) > num.abs(sin_t2), num.arccos(z1/sin_t1)*2., num.arcsin(z2/sin_t2)*2.) lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.) return lat, lon
[docs]def latlon_to_ne(*args): '''Relative cartesian coordinates with respect to a reference location. For two locations, a reference location A and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`pyrocko.orthodrome.azimuth' and :py:func:`pyrocko.orthodrome.distance_accurate50m` .. math:: D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \quad \ \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \ \\mathrm{)}\\\\[0.3cm] n &= D_{AB} \cdot \\cos( \\frac{\pi }{180} \ \\varphi_{\\mathrm{azi},AB} )\\\\ e &= D_{AB} \cdot \\sin( \\frac{\pi }{180} \\varphi_{\\mathrm{azi},AB}) :param refloc: Location reference point :type refloc: :py:class:`pyrocko.orthodrome.Loc` :param loc: Location of interest :type loc: :py:class:`pyrocko.orthodrome.Loc` :return: Northing and easting from refloc to location :rtype: tuple, float ''' azi = azimuth(*args) dist = distance_accurate50m(*args) n, e = math.cos(azi*d2r)*dist, math.sin(azi*d2r)*dist return n, e
[docs]def latlon_to_ne_numpy(lat0, lon0, lat, lon): '''Relative cartesian coordinates with respect to a reference location. For two locations, a reference location (``lat0``, ``lon0``) and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`azimuth` and :py:func:`distance_accurate50m`. :param lat0: reference location latitude :param lon0: reference location longitude :param lat: absolute location latitude :param lon: absolute location longitude :return: ``(n, e)``: relative north and east positions :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` Implemented formulations: .. math:: D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \quad \ \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \ \\mathrm{)}\\\\[0.3cm] n &= D_{AB} \cdot \\cos( \\frac{\pi }{180} \\varphi_{ \ \\mathrm{azi},AB} )\\\\ e &= D_{AB} \cdot \\sin( \\frac{\pi }{180} \\varphi_{ \ \\mathrm{azi},AB} ) ''' azi = azimuth_numpy(lat0, lon0, lat, lon) dist = distance_accurate50m_numpy(lat0, lon0, lat, lon) n = num.cos(azi*d2r)*dist e = num.sin(azi*d2r)*dist return n, e
_wgs84 = None
[docs]def get_wgs84(): global _wgs84 if _wgs84 is None: from geographiclib.geodesic import Geodesic _wgs84 = Geodesic.WGS84 return _wgs84
[docs]def amap(n): def wrap(f): if n == 1: def func(*args): it = num.nditer(args + (None,)) for ops in it: ops[-1][...] = f(*ops[:-1]) return it.operands[-1] elif n == 2: def func(*args): it = num.nditer(args + (None, None)) for ops in it: ops[-2][...], ops[-1][...] = f(*ops[:-2]) return it.operands[-2], it.operands[-1] return func return wrap
[docs]@amap(2) def ne_to_latlon2(lat0, lon0, north_m, east_m): wgs84 = get_wgs84() az = num.arctan2(east_m, north_m)*r2d dist = num.sqrt(east_m**2 + north_m**2) x = wgs84.Direct(lat0, lon0, az, dist) return x['lat2'], x['lon2']
[docs]@amap(2) def latlon_to_ne2(lat0, lon0, lat1, lon1): wgs84 = get_wgs84() x = wgs84.Inverse(lat0, lon0, lat1, lon1) dist = x['s12'] az = x['azi1'] n = num.cos(az*d2r)*dist e = num.sin(az*d2r)*dist return n, e
[docs]@amap(1) def distance_accurate15nm(lat1, lon1, lat2, lon2): wgs84 = get_wgs84() return wgs84.Inverse(lat1, lon1, lat2, lon2)['s12']
[docs]def positive_region(region): '''Normalize parameterization of a rectangular geographical region. :param region: ``(west, east, south, north)`` :returns: ``(west, east, south, north)``, where ``west <= east`` and where ``west`` and ``east`` are in the range ``[-180., 180.+360.[`` ''' west, east, south, north = [float(x) for x in region] assert -180. - 360. <= west < 180. assert -180. < east <= 180. + 360. assert -90. <= south < 90. assert -90. < north <= 90. if east < west: east += 360. if west < -180.: west += 360. east += 360. return (west, east, south, north)
[docs]def points_in_region(p, region): ''' Check what points are contained in a rectangular geographical region. :param p: NumPy array of shape ``(N, 2)`` where each row is a ``(lat, lon)`` pair [deg] :param region: ``(west, east, south, north)`` [deg] :returns: NumPy array of shape ``(N)``, type ``bool`` ''' w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[:, 0], p[:, 0] <= n), num.logical_or( num.logical_and(w <= p[:, 1], p[:, 1] <= e), num.logical_and(w-360. <= p[:, 1], p[:, 1] <= e-360.)))
[docs]def point_in_region(p, region): ''' Check if a point is contained in a rectangular geographical region. :param p: ``(lat, lon)`` [deg] :param region: ``(west, east, south, north)`` [deg] :returns: ``bool`` ''' w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[0], p[0] <= n), num.logical_or( num.logical_and(w <= p[1], p[1] <= e), num.logical_and(w-360. <= p[1], p[1] <= e-360.)))
[docs]def radius_to_region(lat, lon, radius): ''' Get a rectangular region which fully contains a given circular region. :param lat,lon: center of circular region [deg] :param radius: radius of circular region [m] :return: rectangular region as ``(east, west, south, north)`` [deg] ''' radius_deg = radius * m2d if radius_deg < 45.: lat_min = max(-90., lat - radius_deg) lat_max = min(90., lat + radius_deg) absmaxlat = max(abs(lat_min), abs(lat_max)) if absmaxlat > 89: lon_min = -180. lon_max = 180. else: lon_min = max( -180. - 360., lon - radius_deg / math.cos(absmaxlat*d2r)) lon_max = min( 180. + 360., lon + radius_deg / math.cos(absmaxlat*d2r)) lon_min, lon_max, lat_min, lat_max = positive_region( (lon_min, lon_max, lat_min, lat_max)) return lon_min, lon_max, lat_min, lat_max else: return None
[docs]def geographic_midpoint(lats, lons, weights=None): '''Calculate geographic midpoints by finding the center of gravity. This method suffers from instabilities if points are centered around the poles. :param lats: array of latitudes :param lons: array of longitudes :param weights: array weighting factors (optional) :type lats: :py:class:`numpy.ndarray`, ``(N)`` :type lons: :py:class:`numpy.ndarray`, ``(N)`` :type weights: :py:class:`numpy.ndarray`, ``(N)`` :return: Latitudes and longitudes of the modpoints :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' if not weights: weights = num.ones(len(lats)) total_weigth = num.sum(weights) weights /= total_weigth lats = lats * d2r lons = lons * d2r x = num.sum(num.cos(lats) * num.cos(lons) * weights) y = num.sum(num.cos(lats) * num.sin(lons) * weights) z = num.sum(num.sin(lats) * weights) lon = num.arctan2(y, x) hyp = num.sqrt(x**2 + y**2) lat = num.arctan2(z, hyp) return lat/d2r, lon/d2r
[docs]def geographic_midpoint_locations(locations, weights=None): coords = num.array([loc.effective_latlon for loc in locations]) return geographic_midpoint(coords[:, 0], coords[:, 1], weights)
[docs]def geodetic_to_ecef(lat, lon, alt): ''' Convert geodetic coordinates to Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates. [#1]_ [#2]_ :param lat: Geodetic latitude in [deg]. :param lon: Geodetic longitude in [deg]. :param alt: Geodetic altitude (height) in [m] (positive for points outside the geoid). :type lat: float :type lon: float :type alt: float :return: ECEF Cartesian coordinates (X, Y, Z) in [m]. :rtype: tuple, float .. [#1] https://en.wikipedia.org/wiki/ECEF .. [#2] https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #From_geodetic_to_ECEF_coordinates ''' f = earth_oblateness a = earthradius_equator e2 = 2*f - f**2 lat, lon = num.radians(lat), num.radians(lon) # Normal (plumb line) N = a / num.sqrt(1.0 - (e2 * num.sin(lat)**2)) X = (N+alt) * num.cos(lat) * num.cos(lon) Y = (N+alt) * num.cos(lat) * num.sin(lon) Z = (N*(1.0-e2) + alt) * num.sin(lat) return (X, Y, Z)
[docs]def ecef_to_geodetic(X, Y, Z): ''' Convert Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates to geodetic coordinates (Ferrari's solution). :param X, Y, Z: Cartesian coordinates in ECEF system in [m]. :type X, Y, Z: float :return: Geodetic coordinates (lat, lon, alt). Latitude and longitude are in [deg] and altitude is in [m] (positive for points outside the geoid). :rtype: tuple, float .. seealso :: https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #The_application_of_Ferrari.27s_solution ''' f = earth_oblateness a = earthradius_equator b = a * (1. - f) e2 = 2.*f - f**2 # usefull a2 = a**2 b2 = b**2 e4 = e2**2 X2 = X**2 Y2 = Y**2 Z2 = Z**2 r = num.sqrt(X2 + Y2) r2 = r**2 e_prime2 = (a2 - b2)/b2 E2 = a2 - b2 F = 54. * b2 * Z2 G = r2 + (1.-e2)*Z2 - (e2*E2) C = (e4 * F * r2) / (G**3) S = cbrt(1. + C + num.sqrt(C**2 + 2.*C)) P = F / (3. * (S + 1./S + 1.)**2 * G**2) Q = num.sqrt(1. + (2.*e4*P)) dum1 = -(P*e2*r) / (1.+Q) dum2 = 0.5 * a2 * (1. + 1./Q) dum3 = (P * (1.-e2) * Z2) / (Q * (1.+Q)) dum4 = 0.5 * P * r2 r0 = dum1 + num.sqrt(dum2 - dum3 - dum4) U = num.sqrt((r - e2*r0)**2 + Z2) V = num.sqrt((r - e2*r0)**2 + (1.-e2)*Z2) Z0 = (b2*Z) / (a*V) alt = U * (1. - (b2 / (a*V))) lat = num.arctan((Z + e_prime2 * Z0)/r) lon = num.arctan2(Y, X) return (lat*r2d, lon*r2d, alt)
[docs]class Farside(Exception): pass
[docs]def latlon_to_xyz(latlons): if latlons.ndim == 1: return latlon_to_xyz(latlons[num.newaxis, :])[0] points = num.zeros((latlons.shape[0], 3)) lats = latlons[:, 0] lons = latlons[:, 1] points[:, 0] = num.cos(lats*d2r) * num.cos(lons*d2r) points[:, 1] = num.cos(lats*d2r) * num.sin(lons*d2r) points[:, 2] = num.sin(lats*d2r) return points
[docs]def xyz_to_latlon(xyz): if xyz.ndim == 1: return xyz_to_latlon(xyz[num.newaxis, :])[0] latlons = num.zeros((xyz.shape[0], 2)) latlons[:, 0] = num.arctan2( xyz[:, 2], num.sqrt(xyz[:, 0]**2 + xyz[:, 1]**2)) * r2d latlons[:, 1] = num.arctan2( xyz[:, 1], xyz[:, 0]) * r2d return latlons
[docs]def rot_to_00(lat, lon): rot0 = euler_to_matrix(0., -90.*d2r, 0.).A rot1 = euler_to_matrix(-d2r*lat, 0., -d2r*lon).A return num.dot(rot0.T, num.dot(rot1, rot0)).T
[docs]def distances3d(a, b): return num.sqrt(num.sum((a-b)**2, axis=a.ndim-1))
[docs]def circulation(points2): return num.sum( (points2[1:, 0] - points2[:-1, 0]) * (points2[1:, 1] + points2[:-1, 1]))
[docs]def stereographic(points): dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5 points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside() points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis] return points_out
[docs]def stereographic_poly(points): dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5 points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside() points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis] if circulation(points_out) >= 0: raise Farside() return points_out
[docs]def contains_point(polygon, point): rot = rot_to_00(point[0], point[1]) points_xyz = latlon_to_xyz(polygon) points_rot = num.dot(rot, points_xyz.T).T groups = spoly_cut([points_rot], axis=0) for group in groups: for points_g in group: try: points2 = stereographic_poly(points_g) p = Path(points2, closed=True) if p.contains_point((0., 0.)): return True except Farside: pass return False
[docs]def contains_points(polygon, points): points_xyz = latlon_to_xyz(points) center_xyz = num.mean(points_xyz, axis=0) assert num.all( distances3d(points_xyz, center_xyz[num.newaxis, :]) < 1.0) lat, lon = xyz_to_latlon(center_xyz) rot = rot_to_00(lat, lon) points_rot_xyz = num.dot(rot, points_xyz.T).T points_rot_pro = stereographic(points_rot_xyz) poly_xyz = latlon_to_xyz(polygon) poly_rot_xyz = num.dot(rot, poly_xyz.T).T groups = spoly_cut([poly_rot_xyz], axis=0) result = num.zeros(points.shape[0], dtype=num.int) for group in groups: for poly_rot_group_xyz in group: try: poly_rot_group_pro = stereographic_poly(poly_rot_group_xyz) result += path_contains_points( poly_rot_group_pro, points_rot_pro) except Farside: pass return result.astype(num.bool)