# Source code for pyrocko.orthodrome

# http://pyrocko.org - GPLv3
#
# The Pyrocko Developers, 21st Century
# ---|P------/S----------~Lg----------

'''
Some basic geodetic functions.
'''

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
m2d = 1./d2m

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

raise_if_slow_path_contains_points = False

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(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=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.)

num.asarray(x, dtype=float) for x in args])

[docs]class Loc(object):
'''
Simple location representation.

:attrib lat: Latitude in [deg].
:attrib lon: Longitude in [deg].
'''
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}}}
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 in [deg] point A.
:param a_lons: Longitudes in [deg] point A.
:param b_lats: Latitudes in [deg] point B.
:param b_lons: Longitudes in [deg] 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 in [deg] point A.
:param a_lons: Longitudes in [deg] point A.
:param b_lats: Latitudes in [deg] point B.
:param b_lons: Longitudes in [deg] 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 [deg].
: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):
'''
Azimuth and backazimuth from location A towards B and back.

:returns: Azimuth in [deg] from A to B, back azimuth in [deg] from B to A.
:rtype: tuple[float, float]
'''

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'):
'''
Azimuth and backazimuth from location A towards B and back.

Arguments are given as :py:class:numpy.ndarray.

:param a_lats: Latitude(s) in [deg] of point A.
:type a_lats: :py:class:numpy.ndarray
:param a_lons: Longitude(s) in [deg] of point A.
:type a_lons: :py:class:numpy.ndarray
:param b_lats: Latitude(s) in [deg] of point B.
:type b_lats: :py:class:numpy.ndarray
:param b_lons: Longitude(s) in [deg] of point B.
:type b_lons: :py:class:numpy.ndarray

:returns: Azimuth(s) in [deg] from A to B,
back azimuth(s) in [deg] from B to A.
:rtype: :py:class:numpy.ndarray, :py:class:numpy.ndarray
'''

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 in [deg] point A.
:param a_lons: Longitudes in [deg] point A.
:param b_lats: Latitudes in [deg] point B.
:param b_lons: Longitudes in [deg] 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 [deg], distances in [deg].
: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}
G = \\frac{\\pi}{180}
l = \\frac{\\pi}{180}
\\\\[0.5cm]
S = \\sin^2(G) \\cdot \\cos^2(l) +
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 [m].
: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
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}
G_i = \\frac{\\pi}{180}
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) +
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 in [deg] point A.
:param a_lons: Longitudes in [deg] point A.
:param b_lats: Latitudes in [deg] point B.
:param b_lons: Longitudes in [deg] 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 [m].
: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

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 in [deg].
:param lon0: Longitude origin of the cartesian coordinate system in [deg].
:param north_m: Northing distances from origin in [m].
:param east_m: Easting distances from origin in [m].
: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 in [deg].
:rtype: :py:class:numpy.ndarray, (2xN)

'''

gamma = num.arctan2(east_m, north_m)

[docs]def azidist_to_latlon(lat0, lon0, azimuth_deg, distance_deg):
'''
Absolute latitudes and longitudes are calculated from relative changes.

Convenience wrapper to :py:func:azidist_to_latlon_rad with azimuth and
distance given in degrees.

:param lat0: Latitude origin of the cartesian coordinate system in [deg].
:type lat0: float
:param lon0: Longitude origin of the cartesian coordinate system in [deg].
:type lon0: float
:param azimuth_deg: Azimuth from origin in [deg].
:type azimuth_deg: :py:class:numpy.ndarray, (N)
:param distance_deg: Distances from origin in [deg].
:type distance_deg: :py:class:numpy.ndarray, (N)

:return: Array with latitudes and longitudes in [deg].
:rtype: :py:class:numpy.ndarray, (2xN)
'''

lat0, lon0, azimuth_deg/180.*num.pi, distance_deg/180.*num.pi)

'''
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 in [deg].
:param lon0: Longitude origin of the cartesian coordinate system in [deg].
: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 in [deg].
:rtype: :py:class:numpy.ndarray, (2xN)
'''

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 in [deg].
:param lon0: Longitude origin of the cartesian coordinate system in [deg].
:param north_m: Northing distances from origin in [m].
:param east_m: Easting distances from origin in [m].
: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 in [deg].
:rtype: :py:class:numpy.ndarray, (2xN)
'''

b = math.pi/2.-lat0*d2r

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 in [m].
:rtype: tuple[float, 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: Latitude of the reference location in [deg].
:param lon0: Longitude of the reference location in [deg].
:param lat: Latitude of the absolute location in [deg].
:param lon: Longitude of the absolute location in [deg].

:return: (n, e): relative north and east positions in [m].
: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) in [deg].
:type region: :py:class:tuple of :py:class:float

:returns: (west, east, south, north), where west <= east and
where west and east are in the range [-180., 180.+360.].
:rtype: :py:class:tuple of :py:class:float
'''
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: (lat, lon) pairs in [deg].
:type p: :py:class:numpy.ndarray (N, 2)
:param region: (west, east, south, north) region boundaries in [deg].
:type region: :py:class:tuple of :py:class:float

:returns: Mask, returning True for each point within the region.
:rtype: :py:class:numpy.ndarray of bool, shape (N)
'''

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) in [deg].
:type p: :py:class:tuple of :py:class:float
:param region: (west, east, south, north) region boundaries in [deg].
:type region: :py:class:tuple of :py:class:float

:returns: True, if point is in region, else False.
:rtype: 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.)))

'''
Get a rectangular region which fully contains a given circular region.

:param lat: Latitude of the center point of circular region in [deg].
:type lat: float
:param lon: Longitude of the center point of circular region in [deg].
:type lon: float

:returns: Rectangular region as (east, west, south, north) in [deg] or
None.
:rtype: tuple[float, float, float, float]
'''
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_max = min(
180. + 360.,

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(

'''
Calculate geographic midpoints by finding the center of gravity.

This method suffers from instabilities if points are centered around the
poles.

:param lats: Latitudes in [deg].
:param lons: Longitudes in [deg].
:param weights: Weighting factors.
:param depths: Depths in [m].
:type lats: :py:class:numpy.ndarray, (N)
:type lons: :py:class:numpy.ndarray, (N)
:type weights: optional, :py:class:numpy.ndarray, (N)
:type depths: optional, :py:class:numpy.ndarray, (N)

:return: Latitudes and longitudes of the midpoint in [deg] (and depth [m]
if depths are given).
:rtype: (lat, lon) or (lat, lon, depth)
'''
if not weights:
weights = num.ones(len(lats))

total_weigth = num.sum(weights)
weights /= total_weigth
lats = lats * d2r
lons = lons * d2r
if depths is not None:
else:

x = num.sum(num.cos(lats) * num.cos(lons) * weights * radii)
y = num.sum(num.cos(lats) * num.sin(lons) * weights * radii)
z = num.sum(num.sin(lats) * weights * radii)

lon = num.arctan2(y, x)
hyp = num.sqrt(x**2 + y**2)
lat = num.arctan2(z, hyp)

if depths is None:
return lat/d2r, lon/d2r
else:
return lat/d2r, lon/d2r, depth

def geographic_midpoint_locations(
locations, weights=None, include_depth=False):

if not include_depth:
coords = num.array([loc.effective_latlon
for loc in locations])
return geographic_midpoint(
coords[:, 0], coords[:, 1], weights)
else:
coords = num.array([loc.effective_latlon + (loc.depth,)
for loc in locations])
return geographic_midpoint(
coords[:, 0], coords[:, 1], weights=weights, depths=coords[:, 2])

[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, float, 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
e2 = 2*f - f**2

# 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: ECEF X coordinate [m].
:type X: float
:param Y: ECEF Y coordinate [m].
:type Y: float
:param Z: ECEF Z coordinate [m].
:type 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: :py:class:tuple of :py:class:float

.. seealso ::
https://en.wikipedia.org/wiki/Geographic_coordinate_conversion
#The_application_of_Ferrari.27s_solution
'''
f = earth_oblateness
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)

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.)
rot1 = euler_to_matrix(-d2r*lat, 0., -d2r*lon)
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.

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.

: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

:rtype: :py:class:numpy.ndarray of shape (N,).
'''

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]:

center_xyz = num.array([ix, iy, iz], dtype=float)

lat, lon = xyz_to_latlon(center_xyz)
rot = rot_to_00(lat, lon)

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:
poly_rot_group_pro, points_rot_pro)
else:
poly_rot_group_pro, points_rot_pro)

return result.astype(bool)

[docs]def contains_point(polygon, point):
'''
Test if point is inside polygon on a sphere.

Convenience wrapper to :py:func:contains_points to test a single point.

: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].
:type point: :py:class:tuple of :py:class:float

:returns: True, if point is located within polygon, else False.
:rtype: bool
'''

return bool(
contains_points(polygon, num.asarray(point)[num.newaxis, :])[0])