Source code for pyart.core.transforms

"""
Transformations between coordinate systems. Routines for converting between
Cartesian/Cartographic (x, y, z), Geographic (latitude, longitude, altitude)
and antenna (azimuth, elevation, range) coordinate systems.

"""

import warnings

import numpy as np

try:
    import pyproj

    _PYPROJ_AVAILABLE = True
except ImportError:
    _PYPROJ_AVAILABLE = False

from ..exceptions import MissingOptionalDependency

PI = np.pi


[docs]def antenna_to_cartesian(ranges, azimuths, elevations): """ Return Cartesian coordinates from antenna coordinates. Parameters ---------- ranges : array Distances to the center of the radar gates (bins) in kilometers. azimuths : array Azimuth angle of the radar in degrees. elevations : array Elevation angle of the radar in degrees. Returns ------- x, y, z : array Cartesian coordinates in meters from the radar. Notes ----- The calculation for Cartesian coordinate is adapted from equations 2.28(b) and 2.28(c) of Doviak and Zrnic [1]_ assuming a standard atmosphere (4/3 Earth's radius model). .. math:: z = \\sqrt{r^2+R^2+2*r*R*sin(\\theta_e)} - R s = R * arcsin(\\frac{r*cos(\\theta_e)}{R+z}) x = s * sin(\\theta_a) y = s * cos(\\theta_a) Where r is the distance from the radar to the center of the gate, :math:`\\theta_a` is the azimuth angle, :math:`\\theta_e` is the elevation angle, s is the arc length, and R is the effective radius of the earth, taken to be 4/3 the mean radius of earth (6371 km). References ---------- .. [1] Doviak and Zrnic, Doppler Radar and Weather Observations, Second Edition, 1993, p. 21. """ theta_e = np.deg2rad(elevations) # elevation angle in radians. theta_a = np.deg2rad(azimuths) # azimuth angle in radians. R = 6371.0 * 1000.0 * 4.0 / 3.0 # effective radius of earth in meters. r = ranges * 1000.0 # distances to gates in meters. z = (r**2 + R**2 + 2.0 * r * R * np.sin(theta_e)) ** 0.5 - R s = R * np.arcsin(r * np.cos(theta_e) / (R + z)) # arc length in m. x = s * np.sin(theta_a) y = s * np.cos(theta_a) return x, y, z
[docs]def cartesian_to_antenna(x, y, z): """ Returns antenna coordinates from Cartesian coordinates. Parameters ---------- x, y, z : array Cartesian coordinates in meters from the radar. Returns ------- ranges : array Distances to the center of the radar gates (bins) in m. azimuths : array Azimuth angle of the radar in degrees. [-180., 180] elevations : array Elevation angle of the radar in degrees. """ ranges = np.sqrt(x**2.0 + y**2.0 + z**2.0) elevations = np.rad2deg(np.arctan(z / np.sqrt(x**2.0 + y**2.0))) azimuths = np.rad2deg(np.arctan2(x, y)) # [-180, 180] azimuths[azimuths < 0.0] += 360.0 # [0, 360] return ranges, azimuths, elevations
[docs]def antenna_vectors_to_cartesian(ranges, azimuths, elevations, edges=False): """ Calculate Cartesian coordinate for gates from antenna coordinate vectors. Calculates the Cartesian coordinates for the gate centers or edges for all gates from antenna coordinate vectors assuming a standard atmosphere (4/3 Earth's radius model). See :py:func:`pyart.util.antenna_to_cartesian` for details. Parameters ---------- ranges : array, 1D. Distances to the center of the radar gates (bins) in meters. azimuths : array, 1D. Azimuth angles of the rays in degrees. elevations : array, 1D. Elevation angles of the rays in degrees. edges : bool, optional True to calculate the coordinates of the gate edges by interpolating between gates and extrapolating at the boundaries. False to calculate the gate centers. Returns ------- x, y, z : array, 2D Cartesian coordinates in meters from the center of the radar to the gate centers or edges. """ if edges: if len(ranges) != 1: ranges = _interpolate_range_edges(ranges) if len(elevations) != 1: elevations = _interpolate_elevation_edges(elevations) if len(azimuths) != 1: azimuths = _interpolate_azimuth_edges(azimuths) rg, azg = np.meshgrid(ranges, azimuths) rg, eleg = np.meshgrid(ranges, elevations) return antenna_to_cartesian(rg / 1000.0, azg, eleg)
def _interpolate_range_edges(ranges): """Interpolate the edges of the range gates from their centers.""" edges = np.empty((ranges.shape[0] + 1,), dtype=ranges.dtype) edges[1:-1] = (ranges[:-1] + ranges[1:]) / 2.0 edges[0] = ranges[0] - (ranges[1] - ranges[0]) / 2.0 edges[-1] = ranges[-1] - (ranges[-2] - ranges[-1]) / 2.0 edges[edges < 0] = 0 # do not allow range to become negative return edges def _interpolate_elevation_edges(elevations): """Interpolate the edges of the elevation angles from their centers.""" edges = np.empty((elevations.shape[0] + 1,), dtype=elevations.dtype) edges[1:-1] = (elevations[:-1] + elevations[1:]) / 2.0 edges[0] = elevations[0] - (elevations[1] - elevations[0]) / 2.0 edges[-1] = elevations[-1] - (elevations[-2] - elevations[-1]) / 2.0 edges[edges > 180] = 180.0 # prevent angles from going below horizon edges[edges < 0] = 0.0 return edges def _interpolate_azimuth_edges(azimuths): """Interpolate the edges of the azimuth angles from their centers.""" edges = np.empty((azimuths.shape[0] + 1,), dtype=azimuths.dtype) # perform interpolation and extrapolation in complex plane to # account for periodic nature of azimuth angle. azimuths = np.exp(1.0j * np.deg2rad(azimuths)) edges[1:-1] = np.angle(azimuths[1:] + azimuths[:-1], deg=True) half_angle = _half_angle_complex(azimuths[0], azimuths[1]) edges[0] = (np.angle(azimuths[0], deg=True) - half_angle) % 360.0 half_angle = _half_angle_complex(azimuths[-1], azimuths[-2]) edges[-1] = (np.angle(azimuths[-1], deg=True) + half_angle) % 360.0 edges[edges < 0] += 360 # range from [-180, 180] to [0, 360] return edges def _half_angle_complex(complex_angle1, complex_angle2): """ Return half the angle between complex numbers on the unit circle. Parameters ---------- complex_angle1, complex_angle2 : complex Complex numbers representing unit vectors on the unit circle Returns ------- half_angle : float Half the angle between the unit vectors in degrees. """ dot_product = np.real(complex_angle1 * np.conj(complex_angle2)) if dot_product > 1: warnings.warn("dot_product is larger than one.") dot_product = 1.0 full_angle_rad = np.arccos(dot_product) half_angle_rad = full_angle_rad / 2.0 half_angle_deg = np.rad2deg(half_angle_rad) return half_angle_deg def _interpolate_axes_edges(axes): """Interpolate the edges of the axes gates from their centers.""" edges = np.empty((axes.shape[0] + 1,), dtype=axes.dtype) edges[1:-1] = (axes[:-1] + axes[1:]) / 2.0 edges[0] = axes[0] - (axes[1] - axes[0]) / 2.0 edges[-1] = axes[-1] - (axes[-2] - axes[-1]) / 2.0 return edges def antenna_to_cartesian_track_relative(ranges, rot, roll, drift, tilt, pitch): """ Calculate track-relative Cartesian coordinates from radar coordinates. Parameters ---------- ranges : array Distances to the center of the radar gates (bins) in kilometers. rot : array Rotation angle of the radar in degrees. roll : array Roll angle of the radar in degrees. drift : array Drift angle of the radar in degrees. tilt : array Tilt angle of the radar in degrees. pitch : array Pitch angle of the radar in degrees. Returns ------- x, y, z : array Cartesian coordinates in meters from the radar. Notes ----- Project native (polar) coordinate radar sweep data onto track-relative Cartesian coordinate grid. References ---------- .. [1] Lee et al. (1994) Journal of Atmospheric and Oceanic Technology. """ rot = np.radians(rot) # rotation angle in radians. roll = np.radians(roll) # roll angle in radians. drift = np.radians(drift) # drift angle in radians. tilt = np.radians(tilt) # tilt angle in radians. pitch = np.radians(pitch) # pitch angle in radians. r = ranges * 1000.0 # distances to gates in meters. x = r * ( np.cos(rot + roll) * np.sin(drift) * np.cos(tilt) * np.sin(pitch) + np.cos(drift) * np.sin(rot + roll) * np.cos(tilt) - np.sin(drift) * np.cos(pitch) * np.sin(tilt) ) y = r * ( -1.0 * np.cos(rot + roll) * np.cos(drift) * np.cos(tilt) * np.sin(pitch) + np.sin(drift) * np.sin(rot + roll) * np.cos(tilt) + np.cos(drift) * np.cos(pitch) * np.sin(tilt) ) z = r * np.cos(pitch) * np.cos(tilt) * np.cos(rot + roll) + np.sin(pitch) * np.sin( tilt ) return x, y, z def antenna_to_cartesian_earth_relative(ranges, rot, roll, heading, tilt, pitch): """ Calculate earth-relative Cartesian coordinates from radar coordinates Parameters ---------- ranges : array Distances to the center of the radar gates (bins) in kilometers. rot : array Rotation angle of the radar in degrees. roll : array Roll angle of the radar in degrees. heading : array Heading (compass) angle of the radar in degrees clockwise from north. tilt : array Tilt angle of the radar in degrees. pitch : array Pitch angle of the radar in degrees. Returns ------- x, y, z : array Cartesian coordinates in meters from the radar. Notes ----- Project native (polar) coordinate radar sweep data onto earth-relative Cartesian coordinate grid. References ---------- .. [1] Lee et al. (1994) Journal of Atmospheric and Oceanic Technology. """ rot = np.radians(rot) # rotation angle in radians. roll = np.radians(roll) # roll angle in radians. heading = np.radians(heading) # drift angle in radians. tilt = np.radians(tilt) # tilt angle in radians. pitch = np.radians(pitch) # pitch angle in radians. r = ranges * 1000.0 # distances to gates in meters. x = r * ( -1.0 * np.cos(rot + roll) * np.sin(heading) * np.cos(tilt) * np.sin(pitch) + np.cos(heading) * np.sin(rot + roll) * np.cos(tilt) + np.sin(heading) * np.cos(pitch) * np.sin(tilt) ) y = r * ( -1.0 * np.cos(rot + roll) * np.cos(heading) * np.cos(tilt) * np.sin(pitch) - np.sin(heading) * np.sin(rot + roll) * np.cos(tilt) + np.cos(heading) * np.cos(pitch) * np.sin(tilt) ) z = r * np.cos(pitch) * np.cos(tilt) * np.cos(rot + roll) + np.sin(pitch) * np.sin( tilt ) return x, y, z def antenna_to_cartesian_aircraft_relative(ranges, rot, tilt): """ Calculate aircraft-relative Cartesian coordinates from radar coordinates. Parameters ---------- ranges : array Distances to the center of the radar gates (bins) in kilometers. rot : array Rotation angle of the radar in degrees. tilt : array Tilt angle of the radar in degrees. Returns ------- X, Y, Z : array Cartesian coordinates in meters from the radar. Notes ----- Project native (polar) coordinate radar sweep data onto earth-relative Cartesian coordinate grid. References ---------- .. [1] Lee et al. (1994) Journal of Atmospheric and Oceanic Technology. """ rot = np.radians(rot) # rotation angle in radians. tilt = np.radians(tilt) # tilt angle in radians. r = ranges * 1000.0 # distances to gates in meters. x = r * np.cos(tilt) * np.sin(rot) y = r * np.sin(tilt) z = r * np.cos(rot) * np.cos(tilt) return x, y, z
[docs]def geographic_to_cartesian(lon, lat, projparams): """ Geographic to Cartesian coordinate transform. Transform a set of Geographic coordinate (lat, lon) to a Cartesian/Cartographic coordinate (x, y) using pyproj or a build in Azimuthal equidistant projection. Parameters ---------- lon, lat : array-like Geographic coordinates in degrees. projparams : dict or str Projection parameters passed to pyproj.Proj. If this parameter is a dictionary with a 'proj' key equal to 'pyart_aeqd' then a azimuthal equidistant projection will be used that is native to Py-ART and does not require pyproj to be installed. In this case a non-default value of R can be specified by setting the 'R' key to the desired value. Returns ------- x, y : array-like Cartesian coordinates in meters unless projparams defines a value for R in different units. """ if isinstance(projparams, dict) and projparams.get("proj") == "pyart_aeqd": # Use Py-ART's Azimuthal equidistance projection lon_0 = projparams["lon_0"] lat_0 = projparams["lat_0"] if "R" in projparams: R = projparams["R"] x, y = geographic_to_cartesian_aeqd(lon, lat, lon_0, lat_0, R) else: x, y = geographic_to_cartesian_aeqd(lon, lat, lon_0, lat_0) else: # Use pyproj for the projection # check that pyproj is available if not _PYPROJ_AVAILABLE: raise MissingOptionalDependency( "PyProj is required to use geographic_to_cartesian " "with a projection other than pyart_aeqd but it is not " "installed" ) proj = pyproj.Proj(projparams) x, y = proj(lon, lat, inverse=False) return x, y
[docs]def geographic_to_cartesian_aeqd(lon, lat, lon_0, lat_0, R=6370997.0): """ Azimuthal equidistant geographic to Cartesian coordinate transform. Transform a set of geographic coordinates (lat, lon) to Cartesian/Cartographic coordinates (x, y) using a azimuthal equidistant map projection [1]_. .. math:: x = R * k * \\cos(lat) * \\sin(lon - lon_0) y = R * k * [\\cos(lat_0) * \\sin(lat) - \\sin(lat_0) * \\cos(lat) * \\cos(lon - lon_0)] k = c / \\sin(c) c = \\arccos(\\sin(lat_0) * \\sin(lat) + \\cos(lat_0) * \\cos(lat) * \\cos(lon - lon_0)) Where x, y are the Cartesian position from the center of projection; lat, lon the corresponding latitude and longitude; lat_0, lon_0 are the latitude and longitude of the center of the projection; R is the radius of the earth (defaults to ~6371 km). Parameters ---------- lon, lat : array-like Longitude and latitude coordinates in degrees. lon_0, lat_0 : float Longitude and latitude, in degrees, of the center of the projection. R : float, optional Earth radius in the same units as x and y. The default value is in units of meters. Returns ------- x, y : array Cartesian coordinates in the same units as R, typically meters. References ---------- .. [1] Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395, 1987, pp. 191-202. """ lon = np.atleast_1d(np.asarray(lon)) lat = np.atleast_1d(np.asarray(lat)) lon_rad = np.deg2rad(lon) lat_rad = np.deg2rad(lat) lat_0_rad = np.deg2rad(lat_0) lon_0_rad = np.deg2rad(lon_0) lon_diff_rad = lon_rad - lon_0_rad # calculate the arccos after ensuring all values in valid domain, [-1, 1] arg_arccos = np.sin(lat_0_rad) * np.sin(lat_rad) + np.cos(lat_0_rad) * np.cos( lat_rad ) * np.cos(lon_diff_rad) arg_arccos[arg_arccos > 1] = 1 arg_arccos[arg_arccos < -1] = -1 c = np.arccos(arg_arccos) with warnings.catch_warnings(): # division by zero may occur here but is properly addressed below so # the warnings can be ignored warnings.simplefilter("ignore", RuntimeWarning) k = c / np.sin(c) # fix cases where k is undefined (c is zero), k should be 1 k[c == 0] = 1 x = R * k * np.cos(lat_rad) * np.sin(lon_diff_rad) y = ( R * k * ( np.cos(lat_0_rad) * np.sin(lat_rad) - np.sin(lat_0_rad) * np.cos(lat_rad) * np.cos(lon_diff_rad) ) ) return x, y
[docs]def cartesian_to_geographic(x, y, projparams): """ Cartesian to Geographic coordinate transform. Transform a set of Cartesian/Cartographic coordinates (x, y) to a geographic coordinate system (lat, lon) using pyproj or a build in Azimuthal equidistant projection. Parameters ---------- x, y : array-like Cartesian coordinates in meters unless R is defined in different units in the projparams parameter. projparams : dict or str Projection parameters passed to pyproj.Proj. If this parameter is a dictionary with a 'proj' key equal to 'pyart_aeqd' then a azimuthal equidistant projection will be used that is native to Py-ART and does not require pyproj to be installed. In this case a non-default value of R can be specified by setting the 'R' key to the desired value. Returns ------- lon, lat : array Longitude and latitude of the Cartesian coordinates in degrees. """ if isinstance(projparams, dict) and projparams.get("proj") == "pyart_aeqd": # Use Py-ART's Azimuthal equidistance projection lon_0 = projparams["lon_0"] lat_0 = projparams["lat_0"] if "R" in projparams: R = projparams["R"] lon, lat = cartesian_to_geographic_aeqd(x, y, lon_0, lat_0, R) else: lon, lat = cartesian_to_geographic_aeqd(x, y, lon_0, lat_0) else: # Use pyproj for the projection # check that pyproj is available if not _PYPROJ_AVAILABLE: raise MissingOptionalDependency( "PyProj is required to use cartesian_to_geographic " "with a projection other than pyart_aeqd but it is not " "installed" ) proj = pyproj.Proj(projparams) lon, lat = proj(x, y, inverse=True) return lon, lat
[docs]def cartesian_vectors_to_geographic(x, y, projparams, edges=False): """ Cartesian vectors to Geographic coordinate transform. Transform a set of Cartesian/Cartographic coordinate vectors (x, y) to a geographic coordinate system (lat, lon) using pyproj or a build in Azimuthal equidistant projection finding the coordinates edges in Cartesian space if requested. Parameters ---------- x, y : array 1D. Cartesian coordinate vectors in meters unless R is defined in different units in the projparams parameter. projparams : dict or str Projection parameters passed to pyproj.Proj. If this parameter is a dictionary with a 'proj' key equal to 'pyart_aeqd' then a azimuthal equidistant projection will be used that is native to Py-ART and does not require pyproj to be installed. In this case a non-default value of R can be specified by setting the 'R' key to the desired value. edges : bool, optional True to calculate the coordinates of the geographic edges by interpolating between Cartesian points and extrapolating at the boundaries. False to calculate the coordinate centers. Returns ------- lon, lat : array Longitude and latitude of the Cartesian coordinates in degrees. """ if edges: if len(x) > 1: x = _interpolate_axes_edges(x) if len(y) > 1: y = _interpolate_axes_edges(y) x, y = np.meshgrid(x, y) return cartesian_to_geographic(x, y, projparams)
[docs]def cartesian_to_geographic_aeqd(x, y, lon_0, lat_0, R=6370997.0): """ Azimuthal equidistant Cartesian to geographic coordinate transform. Transform a set of Cartesian/Cartographic coordinates (x, y) to geographic coordinate system (lat, lon) using a azimuthal equidistant map projection [1]_. .. math:: lat = \\arcsin(\\cos(c) * \\sin(lat_0) + (y * \\sin(c) * \\cos(lat_0) / \\rho)) lon = lon_0 + \\arctan2( x * \\sin(c), \\rho * \\cos(lat_0) * \\cos(c) - y * \\sin(lat_0) * \\sin(c)) \\rho = \\sqrt(x^2 + y^2) c = \\rho / R Where x, y are the Cartesian position from the center of projection; lat, lon the corresponding latitude and longitude; lat_0, lon_0 are the latitude and longitude of the center of the projection; R is the radius of the earth (defaults to ~6371 km). lon is adjusted to be between -180 and 180. Parameters ---------- x, y : array-like Cartesian coordinates in the same units as R, typically meters. lon_0, lat_0 : float Longitude and latitude, in degrees, of the center of the projection. R : float, optional Earth radius in the same units as x and y. The default value is in units of meters. Returns ------- lon, lat : array Longitude and latitude of Cartesian coordinates in degrees. References ---------- .. [1] Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395, 1987, pp. 191-202. """ x = np.atleast_1d(np.asarray(x)) y = np.atleast_1d(np.asarray(y)) lat_0_rad = np.deg2rad(lat_0) lon_0_rad = np.deg2rad(lon_0) rho = np.sqrt(x * x + y * y) c = rho / R with warnings.catch_warnings(): # division by zero may occur here but is properly addressed below so # the warnings can be ignored warnings.simplefilter("ignore", RuntimeWarning) lat_rad = np.arcsin( np.cos(c) * np.sin(lat_0_rad) + y * np.sin(c) * np.cos(lat_0_rad) / rho ) lat_deg = np.rad2deg(lat_rad) # fix cases where the distance from the center of the projection is zero lat_deg[rho == 0] = lat_0 x1 = x * np.sin(c) x2 = rho * np.cos(lat_0_rad) * np.cos(c) - y * np.sin(lat_0_rad) * np.sin(c) lon_rad = lon_0_rad + np.arctan2(x1, x2) lon_deg = np.rad2deg(lon_rad) # Longitudes should be from -180 to 180 degrees lon_deg[lon_deg > 180] -= 360.0 lon_deg[lon_deg < -180] += 360.0 return lon_deg, lat_deg