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spilhaus.py
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from typing import Tuple
import numpy.typing as npt
import numpy as np
import pandas as pd
from pyproj import Proj, Transformer
from numpy import sin, cos, tan, arcsin, arctan, arctan2, pi
def from_lonlat_to_spilhaus_xy(
longitude: npt.NDArray,
latitude: npt.NDArray
) -> Tuple[npt.NDArray, npt.NDArray]:
"""
Converts longitude and latitude (degrees N and E) into Spilhaus coordinates
:param longitude: -180 to 180
:param latitude: -90 to 90
:return: Spilhaus x (easting) and y (northing)
"""
# constants (https://github.com/OSGeo/PROJ/issues/1851)
e = np.sqrt(0.00669438)
lat_center_deg = -49.56371678
lon_center_deg = 66.94970198
azimuth_deg = 40.17823482
# parameters derived from constants
lat_center_rad = lat_center_deg * pi / 180
lon_center_rad = lon_center_deg * pi / 180
azimuth_rad = azimuth_deg * pi / 180
conformal_lat_center = -pi / 2 + 2 * arctan(
tan(pi/4 + lat_center_rad/2) *
((1 - e * sin(lat_center_rad)) / (1 + e * sin(lat_center_rad))) ** (e / 2)
)
alpha = -arcsin(cos(conformal_lat_center) * cos(azimuth_rad))
lambda_0 = lon_center_rad + arctan2(tan(azimuth_rad), -sin(conformal_lat_center))
beta = pi + arctan2(-sin(azimuth_rad), -tan(conformal_lat_center))
# coordinates in radians
lon = longitude * pi / 180
lat = latitude * pi / 180
# conformal latitude, in radians
lat_c = -pi / 2 + 2 * arctan(
tan(pi/4 + lat/2) * ((1 - e * sin(lat)) / (1 + e * sin(lat))) ** (e / 2)
)
# transformed lat and lon, in degrees
lat_s = 180 / pi * arcsin(sin(alpha) * sin(lat_c) - cos(alpha) * cos(lat_c) * cos(lon - lambda_0))
lon_s = 180 / pi * (
beta + arctan2(
cos(lat_c) * sin(lon - lambda_0),
(sin(alpha) * cos(lat_c) * cos(lon - lambda_0) + cos(alpha) * sin(lat_c))
)
)
# projects transformed coordinates onto plane (Adams World in a Square II)
p = Proj(proj='adams_ws2')
adams_x, adams_y = p(lon_s, lat_s)
spilhaus_x = -(adams_x + adams_y) / np.sqrt(2)
spilhaus_y = (adams_x - adams_y) / np.sqrt(2)
return spilhaus_x, spilhaus_y
def make_spilhaus_xy_gridpoints(
spilhaus_res: int = 1000
) -> pd.DataFrame:
"""
Creates a data frame of Spilhaus coordinates
:param spilhaus_res: grid resolution
:return: df with columns `x` and `y` (nrow = spilhaus_res ** 2, ncol = 2)
"""
# regular grid of points in Spilhaus map
extreme = 11825474
m = np.linspace(start=-extreme, stop=extreme, num=spilhaus_res)
gr = np.array(np.meshgrid(m, m)).reshape(2, spilhaus_res ** 2).T
spilhaus_df = pd.DataFrame({
'x': gr[:, 0],
'y': gr[:, 1]
})
return spilhaus_df
def prettify_spilhaus_df(
spilhaus_df: pd.DataFrame,
z_lower_bound: float = -np.inf,
z_upper_bound: float = np.inf
) -> pd.DataFrame:
"""
Prettifies a Spilhaus data frame
:param spilhaus_df: raw Spilhaus data frame
:param z_lower_bound: lower cutoff for valid z values
:param z_upper_bound: upper cutoff for valid z values
:return: prettier Spilhaus data frame
"""
spilhaus_df['l'] = (
(spilhaus_df['z'] <= z_lower_bound) |
(spilhaus_df['z'] >= z_upper_bound))
spilhaus_x = np.array(spilhaus_df['x'])
spilhaus_y = np.array(spilhaus_df['y'])
spilhaus_z = np.array(spilhaus_df['z'])
spilhaus_l = np.array(spilhaus_df['l'])
extreme = 11825474
spilhaus_res = int(np.sqrt(len(spilhaus_x)))
# augmented grid points
aug_x = np.concatenate([
spilhaus_x,
spilhaus_x - 2 * extreme,
spilhaus_x + 2 * extreme,
spilhaus_x,
spilhaus_x
])
aug_y = np.concatenate([
spilhaus_y,
spilhaus_y,
spilhaus_y,
spilhaus_y - 2 * extreme,
spilhaus_y + 2 * extreme,
])
aug_z = np.concatenate([
spilhaus_z,
np.fliplr(spilhaus_z.reshape(spilhaus_res, spilhaus_res).T).reshape(-1,),
np.flip(spilhaus_z.reshape(spilhaus_res, spilhaus_res).T, axis=1).reshape(-1,),
np.flip(spilhaus_z.reshape(spilhaus_res, spilhaus_res).T, axis=0).reshape(-1,),
np.flip(spilhaus_z.reshape(spilhaus_res, spilhaus_res).T, axis=0).reshape(-1,),
])
aug_l = np.concatenate([
spilhaus_l,
np.fliplr(spilhaus_l.reshape(spilhaus_res, spilhaus_res).T).reshape(-1,),
np.flip(spilhaus_l.reshape(spilhaus_res, spilhaus_res).T, axis=1).reshape(-1,),
np.flip(spilhaus_l.reshape(spilhaus_res, spilhaus_res).T, axis=0).reshape(-1,),
np.flip(spilhaus_l.reshape(spilhaus_res, spilhaus_res).T, axis=0).reshape(-1,),
])
cutpoint = 1.1 * extreme
keep = ~(
aug_l
| (aug_x < -cutpoint)
| (aug_x > cutpoint)
| (aug_y < -cutpoint)
| (aug_y > cutpoint)
| (aug_y > 1.089e7 - 0.176 * aug_x)
| (aug_y > 1.6e7 + 0.8333 * aug_x)
| (aug_x < -0.984e7 - 0.565 * aug_y)
| (aug_y < -1.378e7 + 0.46 * aug_x)
| (aug_x > 1.274e7 + 0.172 * aug_y)
| (aug_y > 1e7 - 0.5 * aug_x)
| (aug_y > 2.3e7 + aug_x)
| ((aug_y < 0.29e7) & (aug_x < -1.114e7))
| ((aug_y < 0.39e7) & (aug_x < -1.17e7))
| ((aug_y < -1.21e7) & (aug_x > 0.295e7))
| ((aug_y < -1.2e7) & (aug_x > 0.312e7))
| ((aug_y < -1.16e7) & (aug_x > 0.4e7))
| ((aug_y < -1.11e7) & (aug_x > 0.45e7))
)
def prettify_axis(u: npt.NDArray) -> npt.NDArray:
unique_u = np.unique(u)
n = len(unique_u)
res_u = np.median(unique_u[1:n] - unique_u[0:(n - 1)])
return ((u - np.min(u)) / res_u).astype(int)
pretty_spilhaus_df = pd.DataFrame({
'x': prettify_axis(aug_x[keep]),
'y': prettify_axis(aug_y[keep]),
'z': aug_z[keep]
}).drop_duplicates(subset=['x', 'y'])
return pretty_spilhaus_df
def from_spilhaus_xy_to_lonlat(
spilhaus_x,
spilhaus_y
) -> Tuple[npt.NDArray, npt.NDArray]:
"""
Converts Spilhaus coordinates into longitude and latitude (degrees N and E)
:param spilhaus_x: Spilhaus easting
:param spilhaus_y: Spilhaus northing
:return: longitude (-180 to 180) and latitude (-90 to 90)
"""
# constants
e = np.sqrt(0.00669438)
lat_center_deg = -49.56371678
lon_center_deg = 66.94970198
azimuth_deg = 40.17823482
# parameters derived from constants
lat_center_rad = lat_center_deg * pi / 180
lon_center_rad = lon_center_deg * pi / 180
azimuth_rad = azimuth_deg * pi / 180
conformal_lat_center = -pi / 2 + 2 * arctan(
tan(pi/4 + lat_center_rad/2) *
((1 - e * sin(lat_center_rad)) / (1 + e * sin(lat_center_rad))) ** (e / 2)
)
alpha = -arcsin(cos(conformal_lat_center) * cos(azimuth_rad))
lambda_0 = lon_center_rad + arctan2(tan(azimuth_rad), -sin(conformal_lat_center))
beta = pi + arctan2(-sin(azimuth_rad), -tan(conformal_lat_center))
# take spilhaus coordinates and compute transformed coords, in degrees
itransformer = Transformer.from_crs({"proj":'adams_ws2'}, 4326, always_xy=True)
adams_x = (spilhaus_y - spilhaus_x) * np.sqrt(2) / 2
adams_y = - (spilhaus_y + spilhaus_x) * np.sqrt(2) / 2
lon_s, lat_s = itransformer.transform(adams_x, adams_y)
#transformed coords in radians
lon_s_rad = lon_s * pi / 180
lat_s_rad = lat_s * pi / 180
# conformal latitude
lat_c = arcsin(sin(alpha) * sin(lat_s_rad) + cos(alpha) * cos(lat_s_rad) * cos(lon_s_rad - beta))
# longitude, in radians
lon = lambda_0 + arctan2(
cos(lat_s_rad) * sin(lon_s_rad - beta),
sin(alpha) * cos(lat_s_rad) * cos(lon_s_rad - beta) - cos(alpha) * sin(lat_s_rad)
)
# latitude (iterative formula from https://mathworld.wolfram.com/ConformalLatitude.html)
lat = lat_c
for i in range(10):
lat = -0.5 * pi + 2 * arctan(
tan(pi / 4 + lat_c / 2) *
((1 + e * sin(lat)) / (1 - e * sin(lat))) ** (e / 2)
)
# coordinates in degrees
longitude = ((lon * 180 / pi + 180) % 360) - 180
latitude = lat * 180 / pi
return longitude, latitude