# http://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------
from __future__ import division, absolute_import
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)
raise_if_slow_path_contains_points = False
[docs]class Slow(Exception):
pass
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
def path_contains_points(verts, points):
if raise_if_slow_path_contains_points:
# used by unit test to skip slow gshhg_example.py
raise Slow()
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.)
def float_array_broadcast(*args):
return num.broadcast_arrays(*[
num.asarray(x, dtype=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):
'''
Clip values of an array.
: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)
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
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 meter
: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=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=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
def get_wgs84():
global _wgs84
if _wgs84 is None:
from geographiclib.geodesic import Geodesic
_wgs84 = Geodesic.WGS84
return _wgs84
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
@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']
@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
@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
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
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
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
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
def distances3d(a, b):
return num.sqrt(num.sum((a-b)**2, axis=a.ndim-1))
def circulation(points2):
return num.sum(
(points2[1:, 0] - points2[:-1, 0])
* (points2[1:, 1] + points2[:-1, 1]))
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
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
def gnomonic_x(points, cutoff=0.01):
points_out = points[:, 1:].copy()
if num.any(points[:, 0] < cutoff):
raise Farside()
factor = 1.0 / points[:, 0]
points_out *= factor[:, num.newaxis]
return points_out
def cneg(i, x):
if i == 1:
return x
else:
return num.logical_not(x)
[docs]def contains_points(polygon, points):
'''
Test which points are inside polygon on a sphere.
:param polygon: Point coordinates defining the polygon [deg].
:type polygon: :py:class:`numpy.ndarray` of shape (N, 2), second index
0=lat, 1=lon
:param points: Coordinates of points to test [deg].
:type points: :py:class:`numpy.ndarray` of shape (N, 2), second index
0=lat, 1=lon
:returns: Boolean mask array.
:rtype: :py:class:`numpy.ndarray` of shape (N,)
The inside of the polygon is defined as the area which is to the left hand
side of an observer walking the polygon line, points in order, on the
sphere. Lines between the polygon points are treated as great circle paths.
The polygon may be arbitrarily complex, as long as it does not have any
crossings or thin parts with zero width. The polygon may contain the poles
and is allowed to wrap around the sphere multiple times.
The algorithm works by consecutive cutting of the polygon into (almost)
hemispheres and subsequent Gnomonic projections to perform the
point-in-polygon tests on a 2D plane.
'''
and_ = num.logical_and
points_xyz = latlon_to_xyz(points)
mask_x = 0. <= points_xyz[:, 0]
mask_y = 0. <= points_xyz[:, 1]
mask_z = 0. <= points_xyz[:, 2]
result = num.zeros(points.shape[0], dtype=int)
for ix in [-1, 1]:
for iy in [-1, 1]:
for iz in [-1, 1]:
mask = and_(
and_(cneg(ix, mask_x), cneg(iy, mask_y)),
cneg(iz, mask_z))
center_xyz = num.array([ix, iy, iz], dtype=float)
lat, lon = xyz_to_latlon(center_xyz)
rot = rot_to_00(lat, lon)
points_rot_xyz = num.dot(rot, points_xyz[mask, :].T).T
points_rot_pro = gnomonic_x(points_rot_xyz)
offset = 0.01
poly_xyz = latlon_to_xyz(polygon)
poly_rot_xyz = num.dot(rot, poly_xyz.T).T
poly_rot_xyz[:, 0] -= offset
groups = spoly_cut([poly_rot_xyz], axis=0)
for poly_rot_group_xyz in groups[1]:
poly_rot_group_xyz[:, 0] += offset
poly_rot_group_pro = gnomonic_x(
poly_rot_group_xyz)
if circulation(poly_rot_group_pro) > 0:
result[mask] += path_contains_points(
poly_rot_group_pro, points_rot_pro)
else:
result[mask] -= path_contains_points(
poly_rot_group_pro, points_rot_pro)
return result.astype(num.bool)
[docs]def contains_point(polygon, point):
'''
Test if point is inside polygon on a sphere.
:param polygon: Point coordinates defining the polygon [deg].
:type polygon: :py:class:`numpy.ndarray` of shape (N, 2), second index
0=lat, 1=lon
:param point: Coordinates ``(lat, lon)`` of point to test [deg].
Convenience wrapper to :py:func:`contains_points` to test a single point.
'''
return bool(
contains_points(polygon, num.asarray(point)[num.newaxis, :])[0])
