geoids.py

# -*- coding: utf-8 -*-

u'''Classes L{GeoidG2012B}, L{GeoidKarney} and L{GeoidPGM} to interpolate
the height of various U{geoid<https://WikiPedia.org/wiki/Geoid>}s at
C{LatLon} locations or separate lat-/longitudes using different
interpolation methods and C{geoid} files.

L{GeoidKarney} is a transcription of I{Charles Karney's} C++ class U{Geoid
<https://GeographicLib.SourceForge.io/html/geoid.html>} to pure Python.

The L{GeoidG2012B} and L{GeoidPGM} interpolators both depend on
U{scipy<https://SciPy.org>} and U{numpy<https://PyPI.org/project/numpy>}
and require those packages to be installed.

In addition, each geoid interpolator needs C{grid knots} to be
(down)loaded from a geoid file, specific to the interpolator, more
details below.  For each interpolator, there are several interpolation
choices, like I{linear}, I{cubic}, etc.

Typical usage is as follows.  First create an interpolator from a
geoid file, also referred to as the C{grid knots}.

C{ginterpolator = GeoidXyz(grid_file, **options)}

Get the interpolated geoid height of C{LatLon} location(s) with

C{h = ginterpolator(ll)}

or

C{h0, h1, h2, ... = ginterpolator(ll0, ll1, ll2, ...)}

or

C{hs = ginterpolator(lls)}  # C{list, tuple, generator, ...}

For separate lat-/longitudes invoke the C{.height} method

C{h = ginterpolator.height(lat, lon)}

or

C{h0, h1, h2, ... = ginterpolator.height(lats, lons)}  # C{list, ...}

Errors from C{scipy} are raised as L{SciPyError}s.  Warnings issued by
C{scipy} can be thrown as L{SciPyWarning} exceptions, provided Python
C{warnings} are filtered accordingly, see L{SciPyWarning}.

@see: Charles Karney's U{GeographicLib<https://GeographicLib.SourceForge.io/
      html/index.html>}, U{Geoid height<https://GeographicLib.SourceForge.io/
      html/geoid.html>} and U{Installing the Geoid datasets<https://
      GeographicLib.SourceForge.io/html/geoid.html#geoidinst>}, U{SciPy
      <https://docs.SciPy.org/doc/scipy/reference/interpolate.html>}
      interpolation U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/
      reference/generated/scipy.interpolate.RectBivariateSpline.html>}
      and U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/
      scipy.interpolate.interp2d.html>} and the functions
      L{elevations.elevation2} and L{elevations.geoidHeight2}.
'''

from pygeodesy.basics import EPS, len2, map1, map2, property_RO
from pygeodesy.datum import Datum, Datums
from pygeodesy.dms import parseDMS2
from pygeodesy.errors import _incompatible, LenError, RangeError, _SciPyIssue
from pygeodesy.fmath import favg, Fdot, fdot, Fhorner, frange
from pygeodesy.heights import _allis2, _ascalar, \
                              _HeightBase, HeightError
from pygeodesy.interns import _COMMA_SPACE_, _cubic_, _E_, _item_cs, \
                              _item_pr, _item_ps, _item_sq, _knots_, \
                              _lat_, _linear_, _lon_, _N_, _n_a_, NN, \
                              _outside_, _S_, _utf_8_, _W_
from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _FOR_DOCS
from pygeodesy.named import LatLon3Tuple, _Named, _NamedTuple, \
                            notOverloaded
from pygeodesy.streprs import attrs, fstr, pairs
from pygeodesy.units import Int_

from math import floor
import os.path as _os_path
from os import SEEK_CUR as _SEEK_CUR, SEEK_SET as _SEEK_SET
from struct import calcsize as _calcsize, unpack as _unpack

try:
    from StringIO import StringIO as _BytesIO  # reads bytes
    _b2str = str  # convert bytes to str

except ImportError:  # Python 3+
    from io import BytesIO as _BytesIO  # PYCHOK expected

    def _b2str(bs):  # used only for egm*.pgm text
        return bs.decode(_utf_8_)

__all__ = _ALL_LAZY.geoids
__version__ = '20.07.08'

# temporarily hold a single instance for each int value
_intCs = {}
_interp2d_ks = {-2: _linear_,
                -3: _cubic_,
                -5: 'quintic'}
_not_supported_ = 'not supported'


class _GeoidBase(_HeightBase):
    '''(INTERNAL) Base class for C{Geoid...}s.
    '''
    _cropped  = None
    _datum    = Datums.WGS84
    _endian   = 'tbd'
    _geoid    = _n_a_
    _hs_y_x   = None  # numpy 2darray, row-major order
    _interp2d = None  # interp2d interpolation
    _kind     = 3  # order for interp2d, RectBivariateSpline
    _knots    = 0  # nlat * nlon
    _mean     = None
#   _name     = NN # _Named
    _nBytes   = 0  # numpy size in bytes, float64
    _pgm      = None
    _sizeB    = 0  # geoid file size in bytes
    _smooth   = 0  # used only for RectBivariateSpline
    _stdev    = None

    _lat_d  = 0.0  # increment, +tive
    _lat_lo = 0.0  # lower lat, south
    _lat_hi = 0.0  # upper lat, noth
    _lon_d  = 0.0  # increment, +tive
    _lon_lo = 0.0  # left lon, west
    _lon_hi = 0.0  # right lon, east
    _lon_of = 0.0  # forward lon offset
    _lon_og = 0.0  # reverse lon offset

    _center     = None  # (lat, lon, height)
    _highest    = None  # (lat, lon, height)
    _lowerleft  = None  # (lat, lon, height)
    _lowerright = None  # (lat, lon, height)
    _lowest     = None  # (lat, lon, height)
    _upperleft  = None  # (lat, lon, height)
    _upperright = None  # (lat, lon, height)

    _yx_hits    = None  # cache hits, ala Karney

    def __init__(self, hs, p):
        '''(INTERNAL) Set up the grid axes, the C{SciPy} interpolator
           and several internal geoid attributes.

           @arg hs: Grid knots with known height (C{numpy 2darray}).
           @arg p: The C{slat, wlon, nlat, nlon, dlat, dlon} and
                   other geoid parameters (C{INTERNAL}).

           @raise GeoidError: Incompatible grid B{C{hs}} shape or
                              invalid B{C{kind}}.

           @raise LenError: Mismatch grid B{C{hs}} axis.

           @raise SciPyError: A C{scipy.interpolate.inter2d} or
                              C{-.RectBivariateSpline} issue.

           @raise SciPyWarning: A C{scipy.interpolate.inter2d} or
                                C{-.RectBivariateSpline} warning as
                                exception.
        '''
        # for 2d scipy.interpolate.interp2d(xs, ys, hs, ...) and
        # scipy.interpolate.RectBivariateSpline(ys, xs, hs, ...)
        # require the shape of hs to be (len(ys), len(xs)), note
        # the different (xs, ys, ...) and (ys, xs, ...) orders
        if (p.nlat, p.nlon) != hs.shape:
            raise GeoidError(shape=hs.shape, txt=_incompatible((p.nlat, p.nlon)))

        # both axes and bounding box
        ys, self._lat_d = self._gaxis2(p.slat, p.dlat, p.nlat, 'lats')
        xs, self._lon_d = self._gaxis2(p.wlon, p.dlon, p.nlon, 'lons')

        bb = ys[0], ys[-1], xs[0], xs[-1] + p.dlon  # fudge lon_hi
        # geoid grids are typically stored in row-major order, some
        # with rows (90..-90) reversed and columns (0..360) wrapped
        # to Easten longitude, 0 <= east < 180 and 180 <= west < 0
        k = self._kind
        if k in _interp2d_ks:
            self._interp2d = self._spi.interp2d(xs, ys, hs, kind=_interp2d_ks[k])
        elif 1 <= k <= 5:
            self._ev = self._spi.RectBivariateSpline(ys, xs, hs, bbox=bb,
                                                                 ky=k, kx=k,
                                                                 s=self._smooth).ev
        else:
            raise GeoidError(kind=k)

        self._hs_y_x = hs  # numpy 2darray, row-major
        self._nBytes = hs.nbytes  # numpy size in bytes
        self._knots  = p.knots  # grid knots
        self._lon_of = float(p.flon)  # forward offset
        self._lon_og = float(p.glon)  # reverse offset
        # shrink the box by 1 unit on every side
        # bb += self._lat_d, -self._lat_d, self._lon_d, -self._lon_d
        self._lat_lo = float(bb[0])
        self._lat_hi = float(bb[1])
        self._lon_lo = float(bb[2] - p.glon)
        self._lon_hi = float(bb[3] - p.glon)

    def __call__(self, *llis):
        '''Interpolate the geoid height for one or several locations.

           @arg llis: The location or locations (C{LatLon}, ... or
                      C{LatLon}s).

           @return: A single interpolated geoid height (C{float}) or
                    a list or tuple of interpolated geoid heights
                    (C{float}s).

           @raise GeoidError: Insufficient number of B{C{llis}} or an
                              invalid B{C{lli}}.

           @raise RangeError: An B{C{lli}} is outside this geoid's lat-
                              or longitude range.

           @raise SciPyError: A C{scipy.interpolate.inter2d} or
                              C{-.RectBivariateSpline} issue.

           @raise SciPyWarning: A C{scipy.interpolate.inter2d} or
                                C{-.RectBivariateSpline} warning as
                                exception.
        '''
        return self._called(llis, True)

    def __repr__(self):
        return self.toStr()

    def __str__(self):
        return _item_pr(self.classname, self.name)

    def _called(self, llis, scipy):
        # handle __call__
        _as, llis = _allis2(llis, Error=GeoidError)
        try:
            hs = []
            for i, lli in enumerate(llis):
                hs.append(self._hGeoid(lli.lat, lli.lon))
            return _as(hs)

        except (GeoidError, RangeError) as x:
            # XXX avoid str(LatLon()) degree symbols
            t = 'lli' if _as is _ascalar else _item_sq(llis=i)
            lli = fstr((lli.lat, lli.lon), strepr=repr)
            raise type(x)(t, lli, txt=str(x))
        except Exception as x:
            if scipy and self.scipy:
                raise _SciPyIssue(x)
            else:
                raise

    def _ev(self, y, x):  # PYCHOK expected
        # only used for .interpolate.interp2d, but
        # overwritten for .RectBivariateSpline,
        # note (y, x) must be flipped!
        return self._interp2d(x, y)

    def _gaxis2(self, lo, d, n, name):
        # build grid axis, hi = lo + (n - 1) * d
        m, a = len2(frange(lo, n, d))
        if m != n:
            raise LenError(self.__class__, grid=m, **{name: n})
        if d < 0:
            d, a = -d, list(reversed(a))
        for i in range(1, m):
            e = a[i] - a[i-1]
            if e < EPS:  # non-increasing axis
                raise GeoidError(_item_sq(name, i), e, txt='non-increasing')
        return self._np.array(a), d

    def _g2ll2(self, lat, lon):  # PYCHOK no cover
        notOverloaded(self, self._g2ll2, lat, lon)

    def _gyx2g2(self, y, x):
        # convert grid (y, x) indices to grid (lat, lon)
        return ((self._lat_lo                + self._lat_d * y),
                (self._lon_lo + self._lon_of + self._lon_d * x))

    def _hGeoid(self, lat, lon):
        out = self.outside(lat, lon)
        if out:
            lli = fstr((lat, lon), strepr=repr)
            raise RangeError('lli', lli, txt='%s on %s' % (_outside_, out))
        return float(self._ev(*self._ll2g2(lat, lon)))

    def _ll2g2(self, lat, lon):  # PYCHOK no cover
        notOverloaded(self, self._ll2g2, lat, lon)

    def _llh3(self, lat, lon):
        r = LatLon3Tuple(lat, lon, self._hGeoid(lat, lon))
        return self._xnamed(r)

    def _llh3LL(self, llh, LatLon):
        return llh if LatLon is None else LatLon(*llh)

    def _llh3minmax(self, highest=True, *unused):
        hs, np = self._hs_y_x, self._np
        # <https://docs.SciPy.org/doc/numpy/reference/generated/
        #         numpy.argmin.html#numpy.argmin>
        arg = self._np.argmax if highest else self._np.argmin
        y, x = np.unravel_index(arg(hs, axis=None), hs.shape)
        return self._g2ll2(*self._gyx2g2(y, x)) + (float(hs[y, x]),)

    def _load(self, g, dtype, n, offset=0):
        # numpy.fromfile, like .frombuffer
        g.seek(offset, _SEEK_SET)
        return self._np.fromfile(g, dtype, n)

    def _open(self, geoid, datum, kind, name, smooth):
        # open the geoid file
        try:
            self._geoid = _os_path.basename(geoid)
            self._sizeB = _os_path.getsize(geoid)
            g = open(geoid, 'rb')
        except (IOError, OSError) as x:
            raise GeoidError(geoid=geoid, txt=str(x))

        if datum and isinstance(datum, Datum):
            self._datum = datum
        self._kind = int(kind)
        if name:
            _HeightBase.name.fset(self, name)  # recursion
        if smooth:
            self._smooth = Int_(smooth, name='smooth', Error=GeoidError, low=0)

        return g

    def _swne(self, crop):
        # crop box to 4-tuple (s, w, n, e)
        try:
            if len(crop) == 2:
                try:  # sw, ne LatLons
                    swne = (crop[0].lat, crop[0].lon,
                            crop[1].lat, crop[1].lon)
                except AttributeError:  # (s, w), (n, e)
                    swne = tuple(crop[0]) + tuple(crop[1])
            else:  # (s, w, n, e)
                swne = crop
            if len(swne) == 4:
                s, w, n, e = map(float, swne)
                if -90 <= s <= (n - 1) <=  89 and \
                  -180 <= w <= (e - 1) <= 179:
                    return s, w, n, e
        except (IndexError, TypeError, ValueError):
            pass
        raise GeoidError(crop=crop)

    def center(self, LatLon=None):
        '''Return the center location and height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the grid
                    center location.
        '''
        if self._center is None:
            self._center = self._llh3(favg(self._lat_lo, self._lat_hi),
                                      favg(self._lon_lo, self._lon_hi))
        return self._llh3LL(self._center, LatLon)

    @property_RO
    def cropped(self):
        '''Is geoid cropped (C{bool} or C{None} if crop not supported).
        '''
        return self._cropped

    @property_RO
    def dtype(self):
        '''Get the grid C{scipy} U{dtype<https://docs.SciPy.org/doc/numpy/
           reference/generated/numpy.ndarray.dtype.html>} (C{numpy.dtype}).
        '''
        return self._hs_y_x.dtype

    @property_RO
    def endian(self):
        '''Get the geoid endianess and U{dtype<https://docs.SciPy.org/
           doc/numpy/reference/generated/numpy.dtype.html>} (C{str}).
        '''
        return self._endian

    def height(self, lats, lons):
        '''Interpolate the geoid height for one or several lat-/longitudes.

           @arg lats: Latitude or latitudes (C{degrees} or C{degrees}s).
           @arg lons: Longitude or longitudes (C{degrees} or C{degrees}s).

           @return: A single interpolated geoid height (C{float}) or a
                    list of interpolated geoid heights (C{float}s).

           @raise GeoidError: Insufficient or non-matching number of
                              B{C{lats}} and B{C{lons}}.

           @raise RangeError: A B{C{lat}} or B{C{lon}} is outside this
                              geoid's lat- or longitude range.

           @raise SciPyError: A C{scipy.interpolate.inter2d} or
                              C{-.RectBivariateSpline} issue.

           @raise SciPyWarning: A C{scipy.interpolate.inter2d} or
                                C{-.RectBivariateSpline} warning as
                                exception.
        '''
        return _HeightBase._height(self, lats, lons, Error=GeoidError)

    def highest(self, LatLon=None, **unused):
        '''Return the location and largest height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the highest
                    grid location.
        '''
        if self._highest is None:
            self._highest = self._llh3minmax(True)
        return self._llh3LL(self._highest, LatLon)

    @property_RO
    def hits(self):
        '''Get the number of cache hits (C{int} or C{None}).
        '''
        return self._yx_hits

    @property_RO
    def kind(self):
        '''Get the interpolator kind and order (C{int}).
        '''
        return self._kind

    @property_RO
    def knots(self):
        '''Get the number of grid knots (C{int}).
        '''
        return self._knots

    def lowerleft(self, LatLon=None):
        '''Return the lower-left location and height of this geoid.

           @kwarg LatLon: Optional class to return the location
                          (C{LatLon}) and height or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the lower-left,
                    SW grid corner.
        '''
        if self._lowerleft is None:
            self._lowerleft = self._llh3(self._lat_lo, self._lon_lo)
        return self._llh3LL(self._lowerleft, LatLon)

    def lowerright(self, LatLon=None):
        '''Return the lower-right location and height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the lower-right,
                    SE grid corner.
        '''
        if self._lowerright is None:
            self._lowerright = self._llh3(self._lat_lo, self._lon_hi)
        return self._llh3LL(self._lowerright, LatLon)

    def lowest(self, LatLon=None, **unused):
        '''Return the location and the lowest height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the lowest
                    grid location.
        '''
        if self._lowest is None:
            self._lowest = self._llh3minmax(False)
        return self._llh3LL(self._lowest, LatLon)

    @property_RO
    def mean(self):
        '''Get the mean of this geoid's heights (C{float}).
        '''
        if self._mean is None:
            self._mean = float(self._np.mean(self._hs_y_x))
        return self._mean

    @property_RO
    def name(self):
        '''Get the name of this geoid (C{str}).
        '''
        return _HeightBase.name.fget(self) or self._geoid  # recursion

    @property_RO
    def nBytes(self):
        '''Get the grid in-memory size in bytes (C{int}).
        '''
        return self._nBytes

    @property_RO
    def numpy(self):
        '''Get the imported C{numpy} version (C{str}).
        '''
        return self._np_v

    def outside(self, lat, lon):
        '''Check whether a location is outside this geoid's
           lat-/longitude or crop range.

           @arg lat: The latitude (C{degrees}).
           @arg lon: The longitude (C{degrees}).

           @return: A 1- or 2-character C{str} if outside or an
                    empty C{str} if inside.
        '''
        return (_S_ if lat < self._lat_lo else
               (_N_ if lat > self._lat_hi else NN)) + \
               (_W_ if lon < self._lon_lo else
               (_E_ if lon > self._lon_hi else NN))

    @property_RO
    def pgm(self):
        '''Get the PGM attributes (C{_PGM} or C{None} if not available/applicable).
        '''
        return self._pgm

    @property_RO
    def scipy(self):
        '''Get the imported C{scipy} version (C{str}).
        '''
        return self._sp_v

    @property_RO
    def sizeB(self):
        '''Get the geoid grid file size in bytes (C{int}).
        '''
        return self._sizeB

    @property_RO
    def smooth(self):
        '''Get the C{RectBivariateSpline} smoothing (C{int}).
        '''
        return self._smooth

    @property_RO
    def stdev(self):
        '''Get the standard deviation of this geoid's heights (C{float}).
        '''
        if self._stdev is None:
            self._stdev = float(self._np.std(self._hs_y_x))
        return self._stdev

    def toStr(self, prec=3, sep=_COMMA_SPACE_):  # PYCHOK signature
        '''This geoid and all geoid attributes as a string.

           @kwarg prec: Optional number of decimal digits (0..9 or
                        C{None} for default).  Trailing zero decimals
                        are stripped for B{C{prec}} values of 1 and above,
                        but kept for negative B{C{prec}} values.
           @kwarg sep: Optional separator (C{str}).

           @return: Geoid name and attributes (C{str}).
        '''
        s = 1 if self.kind < 0 else 2
        t = tuple(_item_ps(m.__name__, fstr(m(), prec=prec)) for m in
                                      (self.lowerleft, self.upperright,
                                       self.center,
                                       self.highest, self.lowest)) + \
            attrs( 'mean', 'stdev',           prec=prec, Nones=False) + \
            attrs(('kind', 'smooth')[:s],     prec=prec, Nones=False) + \
            attrs( 'cropped', 'dtype', 'endian', 'hits', _knots_, 'nBytes',
                   'sizeB', 'scipy', 'numpy', prec=prec, Nones=False)
        return _item_cs(self, sep.join(t))

    def upperleft(self, LatLon=None):
        '''Return the upper-left location and height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the upper-left,
                    NW grid corner.
        '''
        if self._upperleft is None:
            self._upperleft = self._llh3(self._lat_hi, self._lon_lo)
        return self._llh3LL(self._upperleft, LatLon)

    def upperright(self, LatLon=None):
        '''Return the upper-right location and height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the upper-right,
                    NE grid corner.
        '''
        if self._upperright is None:
            self._upperright = self._llh3(self._lat_hi, self._lon_hi)
        return self._llh3LL(self._upperright, LatLon)


class GeoidError(HeightError):
    '''Geoid interpolator C{Geoid...} or interpolation issue.
    '''
    pass


class GeoidG2012B(_GeoidBase):
    '''Geoid height interpolator for U{GEOID12B Model
       <https://www.NGS.NOAA.gov/GEOID/GEOID12B/>} grids U{CONUS
       <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_CONUS.shtml>},
       U{Alaska<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_AK.shtml>},
       U{Hawaii<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_HI.shtml>},
       U{Guam and Northern Mariana Islands
       <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_GMNI.shtml>},
       U{Puerto Rico and U.S. Virgin Islands
       <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_PRVI.shtml>} and
       U{American Samoa<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_AS.shtml>}
       based on C{SciPy} U{RectBivariateSpline<https://docs.SciPy.org/doc/
       scipy/reference/generated/scipy.interpolate.RectBivariateSpline.html>}
       or U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/
       scipy.interpolate.interp2d.html>} interpolation.

       Use any of the binary C{le} (little endian) or C{be} (big endian)
       C{g2012b*.bin} grid files.
    '''
    def __init__(self, g2012b_bin, crop=None, datum=None,  # NAD 83 Ellipsoid
                                   kind=3, name=NN, smooth=0):
        '''New L{GeoidG2012B} interpolator.

           @arg g2012b_bin: A C{GEOID12B} grid file name (C{.bin}).
           @kwarg crop: Optional crop box, not supported (C{None}).
           @kwarg datum: Optional grid datum (C{Datum}), default C{WGS84}.
           @kwarg kind: C{scipy.interpolate} order (C{int}), use 1..5 for
                        U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/
                        reference/generated/scipy.interpolate.RectBivariateSpline.html>},
                        -2 for U{interp2d linear<https://docs.SciPy.org/doc/scipy/
                        reference/generated/scipy.interpolate.interp2d.html>}, -3
                        for C{interp2d cubic} or -5 for C{interp2d quintic}.
           @kwarg name: Optional geoid name (C{str}).
           @kwarg smooth: Smoothing factor for U{RectBivariateSpline
                          <https://docs.SciPy.org/doc/scipy/reference/generated/
                          scipy.interpolate.RectBivariateSpline.html>}
                          only (C{int}).

           @raise GeoidError: G2012B grid file B{C{g2012b_bin}} issue, non-C{None}
                              B{C{crop}} or invalid B{C{kind}} or B{C{smooth}}.

           @raise ImportError: Package C{numpy} or C{scipy} not found
                               or not installed.

           @raise LenError: Grid file B{C{g2012b_bin}} axis mismatch.

           @raise SciPyError: A C{RectBivariateSpline} or C{inter2d} issue.

           @raise SciPyWarning: A C{RectBivariateSpline} or C{inter2d}
                                warning as exception.
        '''
        if crop is not None:
            raise GeoidError(crop=crop, txt=_not_supported_)

        np, _ = self._NumSciPy()
        g = self._open(g2012b_bin, datum, kind, name, smooth)
        try:
            p = _Gpars()
            n = self.sizeB // 4 - 11  # number of f4 heights
            # U{numpy dtype formats are different from Python struct formats
            # <https://docs.SciPy.org/doc/numpy-1.15.0/reference/arrays.dtypes.html>}
            for en_f4 in ('<f4', '>f4'):
                en_ = en_f4[:1]
                # skip 4xf8, get 3xi4
                p.nlat, p.nlon, ien = map(int, self._load(g, en_+'i4', 3, 32))
                if ien == 1:  # correct endian
                    p.knots = p.nlat * p.nlon
                    if p.knots == n and 1 < p.nlat < n \
                                    and 1 < p.nlon < n:
                        self._endian = en_f4
                        break
            else:  # couldn't validate endian
                raise GeoidError('endianess')

            # get the first 4xf8
            p.slat, p.wlon, p.dlat, p.dlon = map(float, self._load(g, en_+'f8', 4))
            # read all f4 heights, ignoring the first 4xf8 and 3xi4
            hs = self._load(g, en_f4, n, 44).reshape(p.nlat, p.nlon)
            p.wlon -= 360  # western-most East longitude to earth (..., lon)
            _GeoidBase.__init__(self, hs, p)

        except Exception as x:
            raise _SciPyIssue(x, 'in', repr(g2012b_bin))
        finally:
            g.close()

    def _g2ll2(self, lat, lon):
        # convert grid (lat, lon) to earth (lat, lon)
        return lat, lon

    def _ll2g2(self, lat, lon):
        # convert earth (lat, lon) to grid (lat, lon)
        return lat, lon

    if _FOR_DOCS:
        __call__ = _GeoidBase.__call__
        height   = _GeoidBase.height


class GeoidHeight5Tuple(_NamedTuple):  # .geoids.py
    '''5-Tuple C{(lat, lon, egm84, egm96, egm2008)} for U{GeoidHeights.dat
       <https://SourceForge.net/projects/geographiclib/files/testdata/>}
       tests with the heights for 3 different EGM grids with C{-90.0 <=
       lat <= 90.0} and C{-180.0 <= lon <= 180.0} degrees (and C{lon}
       converted from the original C{0.0 <= EasterLon <= 360.0}).
    '''
    _Names_ = (_lat_, _lon_, 'egm84', 'egm96', 'egm2008')


def _I(i):
    '''(INTERNAL) Cache a single C{int} constant.
    '''
    return _intCs.setdefault(i, i)  # PYCHOK undefined due to del _intCs


def _T(*cs):
    '''(INTERNAL) Cache a tuple of single C{int} constants.
    '''
    return map1(_I, *cs)


class GeoidKarney(_GeoidBase):
    '''Geoid height interpolator for Charles Karney's U{GeographicLib
       Earth Gravitational Model (EGM)<https://GeographicLib.SourceForge.io/
       html/geoid.html>} geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/
       html/geoid.html#geoidinst>} datasets using bilinear or U{cubic
       <https://dl.ACM.org/citation.cfm?id=368443>} interpolation and U{caching
       <https://GeographicLib.SourceForge.io/html/geoid.html#geoidcache>}
       in pure Python transcribed from Karney's U{C++ class Geoid
       <https://GeographicLib.SourceForge.io/html/geoid.html#geoidinterp>}.

       Use any of the geoid U{egm84-, egm96- or egm2008-*.pgm
       <https://GeographicLib.SourceForge.io/html/geoid.html#geoidinst>}
       datasets.
    '''
    _C0 = (372.0, 240.0, 372.0)  # n, _ and s common denominators
    # matrices c3n_, c3, c3s_, transposed from GeographicLib/Geoid.cpp
    _C3 = ((_T(0,    0,   62,  124,  124,  62,    0,   0,   0,   0,    0,   0),
            _T(0,    0,    0,    0,    0,   0,    0,   0,   0,   0,    0,   0),
         _T(-131,    7,  -31,  -62,  -62, -31,   45, 216, 156, -45,  -55,  -7),
            _T(0,    0,    0,    0,    0,   0,    0,   0,   0,   0,    0,   0),
          _T(138, -138,    0,    0,    0,   0, -183,  33, 153,  -3,   48, -48),  # PYCHOK indent
          _T(144,   42,  -62, -124, -124, -62,   -9,  87,  99,   9,   42, -42),
            _T(0,    0,    0,    0,    0,   0,    0,   0,   0,   0,    0,   0),
            _T(0,    0,    0,    0,    0,   0,   93, -93, -93,  93,    0,   0),
         _T(-102,  102,    0,    0,    0,   0,   18,  12, -12, -18,  -84,  84),
          _T(-31,  -31,   31,   62,   62,  31,    0, -93, -93,   0,   31,  31)),  # PYCHOK indent

           (_T(9,   -9,    9,  186,   54,  -9,   -9,  54, -54,   9,   -9,   9),
          _T(-18,   18,  -88,  -42,  162, -32,    8, -78,  78,  -8,   18, -18),
          _T(-88,    8,  -18,  -42,  -78,  18,   18, 162,  78, -18,  -32,  -8),
            _T(0,    0,   90, -150,   30,  30,   30, -90,  90, -30,    0,   0),
           _T(96,  -96,   96,  -96,  -24,  24,  -96, -24, 144, -24,   24, -24),  # PYCHOK indent
           _T(90,   30,    0, -150,  -90,   0,    0,  30,  90,   0,   30, -30),
            _T(0,    0,  -20,   60,  -60,  20,  -20,  60, -60,  20,    0,   0),
            _T(0,    0,  -60,   60,   60, -60,   60, -60, -60,  60,    0,   0),
          _T(-60,   60,    0,   60,  -60,   0,    0,  60, -60,   0,  -60,  60),
          _T(-20,  -20,    0,   60,   60,   0,    0, -60, -60,   0,   20,  20)),

          (_T(18,  -18,   36,  210,  162, -36,    0,   0,   0,   0,  -18,  18),  # PYCHOK indent
          _T(-36,   36, -165,   45,  141, -21,    0,   0,   0,   0,   36, -36),
         _T(-122,   -2,  -27, -111,  -75,  27,   62, 124, 124,  62,  -64,   2),
            _T(0,    0,   93,  -93,  -93,  93,    0,   0,   0,   0,    0,   0),
          _T(120, -120,  147,  -57, -129,  39,    0,   0,   0,   0,   66, -66),  # PYCHOK indent
          _T(135,   51,   -9, -192, -180,   9,   31,  62,  62,  31,   51, -51),
            _T(0,    0,    0,    0,    0,   0,    0,   0,   0,   0,    0,   0),
            _T(0,    0,  -93,   93,   93, -93,    0,   0,   0,   0,    0,   0),
          _T(-84,   84,   18,   12,  -12, -18,    0,   0,   0,   0, -102, 102),
          _T(-31,  -31,    0,   93,   93,   0,  -31, -62, -62, -31,   31,  31)))

    _BT = (_T(0, 0),  # bilinear 4-tuple [i, j] indices
           _T(1, 0),
           _T(0, 1),
           _T(1, 1))

    _CM = (_T( 0, -1),  # 10x12 cubic matrix [i, j] indices
           _T( 1, -1),
           _T(-1,  0),
           _T( 0,  0),
           _T( 1,  0),
           _T( 2,  0),
           _T(-1,  1),
           _T( 0,  1),
           _T( 1,  1),
           _T( 2,  1),
           _T( 0,  2),
           _T( 1,  2))

    _egm     = None   # open geoid file
    _endian  = '>H'   # struct.unpack 1 ushort (big endian, unsigned short)
    _4endian = '>4H'  # struct.unpack 4 ushorts
    _Rendian = NN     # struct.unpack a row of ushorts
#   _highest = (-8.4,   147.367, 85.839) if egm2008-1.pgm else (
#              (-8.167, 147.25,  85.422) if egm96-5.pgm else
#              (-4.5,   148.75,  81.33)   # egm84-15.pgm)
#   _lowest  = (4.7,   78.767, -106.911) if egm2008-1.pgm else (
#              (4.667, 78.833, -107.043) if egm96-5.pgm else
#              (4.75,  79.25,  -107.34)   # egm84-15.pgm
    _mean    = -1.317  # from egm2008-1, -1.438 egm96-5, -0.855 egm84-15
    _nBytes  = None  # not applicable
    _nterms  = len(_C3[0])  # columns length, number of row
    _smooth  = None  # not applicable
    _stdev   = 29.244  # from egm2008-1, 29.227 egm96-5, 29.183 egm84-15
    _u2B     = _calcsize(_endian)  # pixelsize_ in bytes
    _4u2B    = _calcsize(_4endian)  # 4 pixelsize_s in bytes
    _Ru2B    = 0   # row of pixelsize_s in bytes
    _yxH     = ()  # cache (y, x) indices
    _yxHt    = ()  # cached 4- or 10-tuple for _ev2H resp. _ev3H
    _yx_hits = 0   # cache hits

    def __init__(self, egm_pgm, crop=None, datum=None,  # WGS84
                                kind=3, name=NN, smooth=None):
        '''New L{GeoidKarney} interpolator.

           @arg egm_pgm: An U{EGM geoid dataset<https://GeographicLib.SourceForge.io/
                         html/geoid.html#geoidinst>} file name (C{egm*.pgm}).
           @kwarg crop: Optional box to limit geoid locations, a 4-tuple (C{south,
                        west, north, east}), 2-tuple (C{(south, west), (north,
                        east)}) or 2, in C{degrees90} lat- and C{degrees180}
                        longitudes or a 2-tuple (C{LatLonSW, LatLonNE}) of
                        C{LatLon} instances.
           @kwarg datum: Optional grid datum (C{Datum}), default C{WGS84}.
           @kwarg kind: Interpolation order (C{int}), 2 for C{bilinear} or 3
                        for C{cubic}.
           @kwarg name: Optional geoid name (C{str}).
           @kwarg smooth: Smoothing factor, unsupported (C{None}).

           @raise GeoidError: EGM dataset B{C{egm_pgm}} issue or invalid B{C{crop}},
                              B{C{kind}} or B{C{smooth}}.

           @see: Class L{GeoidPGM} and function L{egmGeoidHeights}.
        '''
        if smooth is not None:
            raise GeoidError(smooth=smooth, txt=_not_supported_)

        if kind in (2,):
            self._evH = self._ev2H
        elif kind not in (3,):
            raise GeoidError(kind=kind)

        self._egm = g = self._open(egm_pgm, datum, kind, name, smooth)
        self._pgm = p = _PGM(g, pgm=egm_pgm, itemsize=self.u2B, sizeB=self.sizeB)

        self._Rendian = self._4endian.replace('4', str(p.nlon))
        self._Ru2B    = _calcsize(self._Rendian)

        self._knots  = p.knots  # grid knots
        self._lon_of = float(p.flon)  # forward offset
        self._lon_og = float(p.glon)  # reverse offset
        # set earth (lat, lon) limits (s, w, n, e)
        self._lat_lo, \
        self._lon_lo, \
        self._lat_hi, \
        self._lon_hi = self._swne(crop if crop else p.crop4)
        self._cropped = True if crop else False

    def __call__(self, *llis):
        '''Interpolate the geoid height for one or several locations.

           @arg llis: The location or locations (C{LatLon}, ... or
                      C{LatLon}s).

           @return: A single interpolated geoid height (C{float}) or
                    a list or tuple of interpolated geoid heights
                    (C{float}s).

           @raise GeoidError: Insufficient number of B{C{llis}} or an
                              invalid B{C{lli}}.

           @raise RangeError: An B{C{lli}} is outside this geoid's lat-
                              or longitude range.
        '''
        return self._called(llis, False)

    def _c0c3v(self, y, x):
        # get the common denominator, the 10x12 cubic matrix and
        # the 12 cubic v-coefficients around geoid index (y, x)
        p = self._pgm
        if 0 < x < (p.nlon - 2) and 0 < y < (p.nlat - 2):
            # read 4x4 ushorts, drop the 4 corners
            g = self._egm
            e = self._4endian
            n = self._4u2B
            R = self._Ru2B

            b = self._seek(y - 1, x - 1)
            v = _unpack(e, g.read(n))[1:3]
            b += R
            g.seek(b, _SEEK_SET)
            v += _unpack(e, g.read(n))
            b += R
            g.seek(b, _SEEK_SET)
            v += _unpack(e, g.read(n))
            b += R
            g.seek(b, _SEEK_SET)
            v += _unpack(e, g.read(n))[1:3]
            j = 1

        else:  # likely some wrapped y and/or x's
            v = self._raws(y, x, GeoidKarney._CM)
            j = 0 if y < 1 else (1 if y < (p.nlat - 2) else 2)

        return GeoidKarney._C0[j], GeoidKarney._C3[j], v

    def _ev(self, lat, lon):  # PYCHOK expected
        # interpolate the geoid height at grid (lat, lon)
        fy, fx = self._g2yx2(lat, lon)
        y, x = int(floor(fy)), int(floor(fx))
        fy -= y
        fx -= x
        H = self._evH(fy, fx, y, x)  # ._ev3H or ._ev2H
        H *= self._pgm.Scale   # H.fmul(self._pgm.Scale)
        H += self._pgm.Offset  # H.fadd(self._pgm.Offset)
        return H.fsum()

    def _ev2H(self, fy, fx, *yx):
        # compute the bilinear 4-tuple and interpolate raw H
        if self._yxH == yx:
            t = self._yxHt
            self._yx_hits += 1
        else:
            y, x = self._yxH = yx
            self._yxHt = t = self._raws(y, x, GeoidKarney._BT)
        v = 1.0, -fx, fx
        H = Fdot(v, t[0], t[0], t[1])  # a
        H -= H * fy  # c = (1 - fy) * a
        H += Fdot(v, t[2], t[2], t[3]) * fy  # c += b * fy
        return H

    def _ev3H(self, fy, fx, *yx):
        # compute the cubic 10-tuple and interpolate raw H
        if self._yxH == yx:
            t = self._yxHt
            self._yx_hits += 1
        else:
            self._yxH = yx
            c0, c3, v = self._c0c3v(*yx)
            t = [fdot(v, *c3[i]) / c0 for i in range(self._nterms)]
            self._yxHt = t = tuple(t)
        # GeographicLib/Geoid.cpp Geoid::height(lat, lon) ...
        # real h = t[0] + fx * (t[1] + fx * (t[3] + fx * t[6])) +
        #                 fy * (t[2] + fx * (t[4] + fx * t[7]) +
        #                 fy * (t[5] + fx *  t[8] + fy * t[9]));
        H = Fdot((1, fx, fy), t[5], t[8], t[9]) * fy
        H = (Fhorner(fx, t[2], t[4], t[7]) + H) * fy
        H += Fhorner(fx, t[0], t[1], t[3], t[6])
        return H

    _evH = _ev3H  # overriden for kind == 2

    def _g2ll2(self, lat, lon):
        # convert grid (lat, lon) to earth (lat, lon), uncropped
        while lon > 180:
            lon -= 360
        return lat, lon

    def _g2yx2(self, lat, lon):
        # convert grid (lat, lon) to grid (y, x) indices
        p = self._pgm
        # note, slat = +90, rlat < 0 makes y >=0
        return ((lat - p.slat) * p.rlat), ((lon - p.wlon) * p.rlon)

    def _gyx2g2(self, y, x):
        # convert grid (y, x) indices to grid (lat, lon)
        p = self._pgm
        return (p.slat + p.dlat * y), (p.wlon + p.dlon * x)

    def _lat2y2(self, lat2):
        # convert earth lat(s) to min and max grid y indices
        ys, m = [], self._pgm.nlat - 1
        for lat in lat2:
            y, _ = self._g2yx2(*self._ll2g2(lat, 0))
            ys.append(max(min(int(y), m), 0))
        return min(ys), max(ys) + 1

    def _ll2g2(self, lat, lon):
        # convert earth (lat, lon) to grid (lat, lon), uncropped
        while lon < 0:
            lon += 360
        return lat, lon

    def _llh3minmax(self, highest=True, *lat2):
        # find highest or lowest, takes 10+ secs for egm2008-1.pgm geoid
        # (Python 2.7.16, macOS 10.13.6 High Sierra, iMac 3 GHz Core i3)
        y = x = 0
        h = self._raw(y, x)
        if highest:
            for j, r in self._raw2(*lat2):
                m = max(r)
                if m > h:
                    h, y, x = m, j, r.index(m)
        else:  # lowest
            for j, r in self._raw2(*lat2):
                m = min(r)
                if m < h:
                    h, y, x = m, j, r.index(m)
        h *= self._pgm.Scale
        h += self._pgm.Offset
        return self._g2ll2(*self._gyx2g2(y, x)) + (h,)

    def _raw(self, y, x):
        # get the ushort geoid height at geoid index (y, x),
        # like GeographicLib/Geoid.hpp real rawval(is, iy)
        p = self._pgm
        if x < 0:
            x += p.nlon
        elif x >= p.nlon:
            x -= p.nlon
        h = p.nlon // 2
        if y < 0:
            y = -y
        elif y >= p.nlat:
            y = (p.nlat - 1) * 2 - y
        else:
            h = 0
        x += h if x < h else -h
        self._seek(y, x)
        h = _unpack(self._endian, self._egm.read(self._u2B))
        return h[0]

    def _raws(self, y, x, ijs):
        # get bilinear 4-tuple or 10x12 cubic matrix
        return tuple(self._raw(y + j, x + i) for i, j in ijs)

    def _raw2(self, *lat2):
        # yield a 2-tuple (y, ushorts) for each row or for
        # the rows between two (or more) earth lat values
        p = self._pgm
        g = self._egm
        e = self._Rendian
        n = self._Ru2B
        # min(lat2) <= lat <= max(lat2) or 0 <= y < p.nlat
        s, t = self._lat2y2(lat2) if lat2 else (0, p.nlat)
        self._seek(s, 0)  # to start of row s
        for y in range(s, t):
            yield y, _unpack(e, g.read(n))

    def _seek(self, y, x):
        # position geoid to grid index (y, x)
        p = self._pgm
        b = p.skip + (y * p.nlon + x) * self._u2B
        self._egm.seek(b, _SEEK_SET)
        return b  # position

    @property_RO
    def dtype(self):
        '''Get the geoid's grid data type (C{str}).
        '''
        return 'ushort'

    def height(self, lats, lons):
        '''Interpolate the geoid height for one or several lat-/longitudes.

           @arg lats: Latitude or latitudes (C{degrees} or C{degrees}s).
           @arg lons: Longitude or longitudes (C{degrees} or C{degrees}s).

           @return: A single interpolated geoid height (C{float}) or a
                    list of interpolated geoid heights (C{float}s).

           @raise GeoidError: Insufficient or non-matching number of
                              B{C{lats}} and B{C{lons}}.

           @raise RangeError: A B{C{lat}} or B{C{lon}} is outside this
                              geoid's lat- or longitude range.
        '''
        return _HeightBase._height(self, lats, lons, Error=GeoidError)

    def highest(self, LatLon=None, full=False):  # PYCHOK full
        '''Return the location and largest height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.
           @kwarg full: Search the full or limited latitude range (C{bool}).

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the highest
                    grid location.
        '''
        if self._highest is None:
            lat2 = () if full or self.cropped else (-12, -4)
            self._highest = self._llh3minmax(True, *lat2)
        return self._llh3LL(self._highest, LatLon)

    def lowest(self, LatLon=None, full=False):  # PYCHOK full
        '''Return the location and the lowest height of this geoid.

           @kwarg LatLon: Optional class to return the location and height
                          (C{LatLon}) or C{None}.
           @kwarg full: Search the full or limited latitude range (C{bool}).

           @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat,
                    lon, height)} otherwise a B{C{LatLon}} instance
                    with the lat-, longitude and height of the lowest
                    grid location.
        '''
        if self._lowest is None:
            lat2 = () if full or self.cropped else (0, 8)
            self._lowest = self._llh3minmax(False, *lat2)
        return self._llh3LL(self._lowest, LatLon)

    @property_RO
    def u2B(self):
        '''Get the PGM itemsize in bytes (C{int}).
        '''
        return self._u2B


class GeoidPGM(_GeoidBase):
    '''Geoid height interpolator for Charles Karney's U{GeographicLib
       Earth Gravitational Model (EGM)<https://GeographicLib.SourceForge.io/
       html/geoid.html>} geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/
       html/geoid.html#geoidinst>} datasets but based on C{SciPy}
       U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/reference/
       generated/scipy.interpolate.RectBivariateSpline.html>} or
       U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/
       scipy.interpolate.interp2d.html>} interpolation.

       Use any of the U{egm84-, egm96- or egm2008-*.pgm
       <https://GeographicLib.SourceForge.io/html/geoid.html#geoidinst>}
       datasets.  However, unless cropped, an entire C{egm*.pgm} dataset
       is loaded into the C{SciPy} U{RectBivariateSpline<https://docs.SciPy.org/
       doc/scipy/reference/generated/scipy.interpolate.RectBivariateSpline.html>}
       or U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/
       scipy.interpolate.interp2d.html>} interpolator and converted from
       2-byte C{int} to 8-byte C{dtype float64}.  Therefore, internal memory
       usage is 4x the U{egm*.pgm<https://GeographicLib.SourceForge.io/html/
       geoid.html#geoidinst>} file size and may exceed the available memory,
       especially with 32-bit Python, see properties C{.nBytes} and C{.sizeB}.
    '''
    _endian = '>u2'
    _u2B    = 0  # np.itemsize

    def __init__(self, egm_pgm, crop=None, datum=None,  # WGS84
                                kind=3, name=NN, smooth=0):
        '''New L{GeoidPGM} interpolator.

           @arg egm_pgm: An U{EGM geoid dataset<https://GeographicLib.SourceForge.io/
                         html/geoid.html#geoidinst>} file name (C{egm*.pgm}).
           @kwarg crop: Optional box to crop B{C{egm_pgm}}, a 4-tuple (C{south, west,
                        north, east}) or 2-tuple (C{(south, west), (north, east)}),
                        in C{degrees90} lat- and C{degrees180} longitudes or a
                        2-tuple (C{LatLonSW, LatLonNE}) of C{LatLon} instances.
           @kwarg datum: Optional grid datum (C{Datum}), default C{WGS84}.
           @kwarg kind: C{scipy.interpolate} order (C{int}), use 1..5 for
                        U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/
                        reference/generated/scipy.interpolate.RectBivariateSpline.html>},
                        -2 for U{interp2d linear<https://docs.SciPy.org/doc/scipy/
                        reference/generated/scipy.interpolate.interp2d.html>}, -3
                        for C{interp2d cubic} or -5 for C{interp2d quintic}.
           @kwarg name: Optional geoid name (C{str}).
           @kwarg smooth: Smoothing factor for U{RectBivariateSpline
                          <https://docs.SciPy.org/doc/scipy/reference/generated/
                          scipy.interpolate.RectBivariateSpline.html>}
                          only (C{int}).

           @raise GeoidError: EGM dataset B{C{egm_pgm}} issue or invalid B{C{crop}},
                              B{C{kind}} or B{C{smooth}}.

           @raise ImportError: Package C{numpy} or C{scipy} not found
                               or not installed.

           @raise LenError: EGM dataset B{C{egm_pgm}} axis mismatch.

           @raise SciPyError: A C{RectBivariateSpline} or C{inter2d} issue.

           @raise SciPyWarning: A C{RectBivariateSpline} or C{inter2d}
                                warning as exception.

           @note: The U{GeographicLib egm*.pgm<https://GeographicLib.SourceForge.io/
                  html/geoid.html#geoidinst>} file sizes are based on a 2-byte
                  C{int} height converted to 8-byte C{dtype float64} for C{scipy}
                  interpolators.  Therefore, internal memory usage is 4 times the
                  C{egm*.pgm} file size and may exceed the available memory,
                  especially with 32-bit Python.  To reduce memory usage, set
                  keyword argument B{C{crop}} to the region of interest.  For example
                  B{crop=(20, -125, 50, -65)} covers the U{conterminous US<https://
                  www.NGS.NOAA.gov/GEOID/GEOID12B/maps/GEOID12B_CONUS_grids.png>}
                  (CONUS), less than 3% of the entire C{egm2008-1.pgm} dataset.

           @see: Class L{GeoidKarney} and function L{egmGeoidHeights}.
        '''
        np, _ = self._NumSciPy()
        self._u2B = np.dtype(self.endian).itemsize

        g = self._open(egm_pgm, datum, kind, name, smooth)
        self._pgm = p = _PGM(g, pgm=egm_pgm, itemsize=self.u2B, sizeB=self.sizeB)
        if crop:
            g = p._cropped(g, abs(kind) + 1, *self._swne(crop))
            self._g2ll2 = self._g2ll2_cropped
            self._ll2g2 = self._ll2g2_cropped
            if map2(int, self.numpy.split('.')[:2]) < (1, 9):
                g = open(g.name, 'rb')  # reopen tempfile for numpy 1.8.0-
            self._cropped = True
        else:
            self._cropped = False

        try:
            # U{numpy dtype formats are different from Python struct formats
            # <https://docs.SciPy.org/doc/numpy-1.15.0/reference/arrays.dtypes.html>}
            # read all heights, skipping the PGM header lines, converted to float
            hs = self._load(g, self.endian, p.knots, p.skip).reshape(p.nlat, p.nlon) * p.Scale
            if p.Offset:  # offset
                hs = p.Offset + hs
            if p.dlat < 0:  # flip the rows
                hs = np.flipud(hs)
            _GeoidBase.__init__(self, hs, p)
        except Exception as x:
            raise _SciPyIssue(x, 'in', repr(egm_pgm))
        finally:
            g.close()

    def _g2ll2(self, lat, lon):
        # convert grid (lat, lon) to earth (lat, lon), uncropped
        while lon > 180:
            lon -= 360
        return lat, lon

    def _g2ll2_cropped(self, lat, lon):
        # convert cropped grid (lat, lon) to earth (lat, lon)
        return lat, lon - self._lon_og

    def _ll2g2(self, lat, lon):
        # convert earth (lat, lon) to grid (lat, lon), uncropped
        while lon < 0:
            lon += 360
        return lat, lon

    def _ll2g2_cropped(self, lat, lon):
        # convert earth (lat, lon) to cropped grid (lat, lon)
        return lat, lon + self._lon_of

    if _FOR_DOCS:
        __call__ = _GeoidBase.__call__
        height   = _GeoidBase.height

    @property_RO
    def u2B(self):
        '''Get the PGM itemsize in bytes (C{int}).
        '''
        return self._u2B


class _Gpars(_Named):
    '''(INTERNAL) Basic geoid parameters.
    '''
    # interpolator parameters
    dlat = 0  # +/- latitude resolution in C{degrees}.
    dlon = 0  # longitude resolution in C{degrees}.
    nlat = 1  # number of latitude knots (C{int}).
    nlon = 0  # number of longitude knots (C{int})
    rlat = 0  # +/- latitude resolution in C{float}, 1 / .dlat.
    rlon = 0  # longitude resolution in C{float}, 1 / .dlon.
    slat = 0  # nothern- or southern most latitude (C{degrees90}).
    wlon = 0  # western-most longitude in Eastern lon (C{degrees360}).

    flon = 0  # forward, earth to grid longitude offset
    glon = 0  # reverse, grid to earth longitude offset

    knots = 0  # number of knots, nlat * nlon (C{int}).
    skip  = 0  # header bytes to skip (C{int}).

    def __repr__(self):
        t = _COMMA_SPACE_.join(pairs((a, getattr(self, a)) for a in
                                             dir(self.__class__) if
                             a[:1].isupper()))
        return _item_cs(self, t)

    def __str__(self):
        return _item_pr(self.classname, self.name)


class _PGM(_Gpars):
    '''(INTERNAL) Parse an C{egm*.pgm} geoid dataset file.

       # Geoid file in PGM format for the GeographicLib::Geoid class
       # Description WGS84 EGM96, 5-minute grid
       # URL https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm96/egm96.html
       # DateTime 2009-08-29 18:45:03
       # MaxBilinearError 0.140
       # RMSBilinearError 0.005
       # MaxCubicError 0.003
       # RMSCubicError 0.001
       # Offset -108
       # Scale 0.003
       # Origin 90N 0E
       # AREA_OR_POINT Point
       # Vertical_Datum WGS84
       <width> <height>
       <pixel>
       ...
    '''
    crop4 = ()  # 4-tuple (C{south, west, north, east}).
    egm   = None
    glon  = 180  # reverse offset, uncropped
#   pgm   = NN   # name
    sizeB = 0
    u2B   = 2  # item size of grid height (C{int}).

    @staticmethod
    def _llstr2floats(latlon):
        # llstr to (lat, lon) floats
        lat, lon = latlon.split()
        return parseDMS2(lat, lon)

    # PGM file attributes, CamelCase but not .istitle()
    AREA_OR_POINT    = str
    DateTime         = str
    Description      = str  # 'WGS84 EGM96, 5-minute grid'
    Geoid            = str  # 'file in PGM format for the GeographicLib::Geoid class'
    MaxBilinearError = float
    MaxCubicError    = float
    Offset           = float
    Origin           = _llstr2floats
    Pixel            = 0
    RMSBilinearError = float
    RMSCubicError    = float
    Scale            = float
    URL              = str  # 'https://Earth-Info.NGA.mil/GandG/wgs84/...'
    Vertical_Datum   = str

    def __init__(self, g, pgm=NN, itemsize=0, sizeB=0):  # MCCABE 22
        '''(INTERNAL) New C{_PGM} parsed C{egm*.pgm} geoid dataset.
        '''
        self.name = pgm  # geoid file name
        if itemsize:
            self._u2B = itemsize
        if sizeB:
            self.sizeB = sizeB

        t = g.readline()  # make sure newline == '\n'
        if t != b'P5\n' or t.strip() != b'P5':
            raise self._Errorf('header %r', t)

        while True:  # read all # Attr ... lines,
            try:  # ignore empty ones or comments
                t = g.readline().strip()
                if t.startswith(b'#'):
                    t = t.lstrip(b'#').lstrip()
                    a, v = map(_b2str, t.split(None, 1))
                    f = getattr(_PGM, a, None)
                    if callable(f) and a[:1].isupper():
                        setattr(self, a, f(v))
                elif t:
                    break
            except (TypeError, ValueError):
                raise self._Errorf('Attr %r', t)
        else:  # should never get here
            raise self._Errorf('header %r', g.tell())

        try:  # must be (even) width and (odd) height
            nlon, nlat = map(int, t.split())
            if nlon < 2 or nlon > (360 * 60) or (nlon & 1) != 0 or \
               nlat < 2 or nlat > (181 * 60) or (nlat & 1) == 0:
                raise ValueError
        except (TypeError, ValueError):
            raise self._Errorf('width height %r', t)

        try:  # must be 16 bit pixel height
            t = g.readline().strip()
            self.Pixel = int(t)
            if not 255 < self.Pixel < 65536:  # >u2 or >H only
                raise ValueError
        except (TypeError, ValueError):
            raise self._Errorf('pixel %r', t)

        for a in dir(_PGM):  # set undefined # Attr ... to None
            if a[:1].isupper() and callable(getattr(self, a)):
                setattr(self, a, None)

        if self.Origin is None:
            raise self._Errorf('Origin %r', self.Origin)
        if self.Offset is None or self.Offset > 0:
            raise self._Errorf('Offset %r', self.Offset)
        if self.Scale is None or self.Scale < EPS:
            raise self._Errorf('Scale %r', self.Scale)

        self.skip = g.tell()
        self.knots = nlat * nlon

        self.nlat, self.nlon = nlat, nlon
        self.slat, self.wlon = self.Origin
        # note, negative .dlat and .rlat since rows
        # are from .slat 90N down in decreasing lat
        self.dlat, self.dlon = 180.0 / (1 - nlat), 360.0 / nlon
        self.rlat, self.rlon = (1 - nlat) / 180.0, nlon / 360.0

        # grid corners in earth (lat, lon), .slat = 90, .dlat < 0
        n = float(self.slat)
        s = n + self.dlat * (nlat - 1)
        w = self.wlon - self.glon
        e = w + self.dlon * nlon
        self.crop4 = s, w, n, e

        n = self.sizeB - self.skip
        if n > 0 and n != (self.knots * self.u2B):
            raise self._Errorf('assert(%s x %s != %s)', nlat, nlon, n)

    def _cropped(self, g, k1, south, west, north, east):  # MCCABE 15
        '''Crop the geoid to (south, west, north, east) box.
        '''
        # flon offset for both west and east
        f = 360 if west < 0 else 0
        # earth (lat, lon) to grid indices (y, x),
        # note y is decreasing, i.e. n < s
        s, w = self._lle2yx2(south, west, f)
        n, e = self._lle2yx2(north, east, f)
        s += 1  # s > n
        e += 1  # e > w

        hi, wi = self.nlat, self.nlon
        # handle special cases
        if (s - n) > hi:
            n, s = 0, hi  # entire lat range
        if (e - w) > wi:
            w, e, f = 0, wi, 180  # entire lon range
        if s == hi and w == n == 0 and e == wi:
            return g  # use entire geoid as-is

        if (e - w) < k1 or (s - n) < (k1 + 1):
            raise self._Errorf('swne %r', (north - south, east - west))

        if e > wi > w:  # wrap around
            # read w..wi and 0..e
            r, p = (wi - w), (e - wi)
        elif e > w:
            r, p = (e - w), 0
        else:
            raise self._Errorf('assert(%s < %s)', w, e)

        # convert to bytes
        r *= self.u2B
        p *= self.u2B
        q = wi * self.u2B  # stride
        # number of rows and cols to skip from
        # the original (.slat, .wlon) origin
        z = self.skip + (n * wi + w) * self.u2B
        # sanity check
        if r < 2 or p < 0 or q < 2 or z < self.skip \
                                   or z > self.sizeB:
            raise self._Errorf('assert%r', (r, p, q, z))

        # can't use _BytesIO since numpy
        # needs .fileno attr in .fromfile
        t, c = 0, self._tmpfile()
        # reading (s - n) rows, forward
        for y in range(n, s):  # PYCHOK y unused
            g.seek(z, _SEEK_SET)
            # Python 2 tmpfile.write returns None
            t += c.write(g.read(r)) or r
            if p:  # wrap around to start of row
                g.seek(-q, _SEEK_CUR)
                # assert(g.tell() == (z - w * self.u2B))
                # Python 2 tmpfile.write returns None
                t += c.write(g.read(p)) or p
            z += q
        c.flush()
        g.close()

        s -= n  # nlat
        e -= w  # nlon
        k = s * e  # knots
        z = k * self.u2B
        if t != z:
            raise self._Errorf('assert(%s != %s) %s', t, z, self)

        # update the _Gpars accordingly, note attributes
        # .dlat, .dlon, .rlat and .rlon remain unchanged
        self.slat += n * self.dlat
        self.wlon += w * self.dlon
        self.nlat = s
        self.nlon = e
        self.flon = self.glon = f

        self.crop4 = south, west, north, east
        self.knots = k
        self.skip  = 0  # no header lines in c

        c.seek(0, _SEEK_SET)
        # c = open(c.name, 'rb')  # reopen for numpy 1.8.0-
        return c

    def _Errorf(self, fmt, *args):  # PYCHOK no cover
        e = self.pgm or ''
        if e:
            e = ' in %r' % (e,)
        return PGMError('%s%s' % ((fmt % args), e))

    def _lle2yx2(self, lat, lon, flon):
        # earth (lat, lon) to grid indices (y, x)
        # with .dlat decreasing from 90N .slat
        lat -= self.slat
        lon += flon - self.wlon
        return (min(self.nlat - 1, max(0, int(lat * self.rlat))),
                                   max(0, int(lon * self.rlon)))

    def _tmpfile(self):
        # create a tmpfile to hold the cropped geoid grid
        try:
            from tempfile import NamedTemporaryFile as tmpfile
        except ImportError:  # Python 2.7.16-
            from os import tmpfile
        t = _os_path.splitext(_os_path.basename(self.pgm))[0]
        f = tmpfile(mode='w+b', prefix=t or 'egm')
        f.seek(0, _SEEK_SET)  # force overwrite
        return f

    @property_RO
    def pgm(self):
        '''Get the geoid file name (C{str}).
        '''
        return self.name


class PGMError(GeoidError):
    '''Issue parsing or cropping an C{egm*.pgm} geoid dataset.
    '''
    pass


def egmGeoidHeights(GeoidHeights_dat):
    '''Generate geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/
       html/geoid.html#geoidinst>} height tests from U{GeoidHeights.dat
       <https://SourceForge.net/projects/geographiclib/files/testdata/>}
       U{Test data for Geoids<https://GeographicLib.SourceForge.io/html/
       geoid.html#testgeoid>}.

       @arg GeoidHeights_dat: The un-gz-ed C{GeoidHeights.dat} file
                              (C{str} or C{file} handle).

       @return: For each test, yield a L{GeoidHeight5Tuple}C{(lat, lon,
                egm84, egm96, egm2008)}.

       @raise GeoidError: Invalid B{C{GeoidHeights_dat}}.

       @note: Function L{egmGeoidHeights} is used to test the geoids
              L{GeoidKarney} and L{GeoidPGM}, see PyGeodesy module
              C{test/testGeoids.py}.
    '''
    dat = GeoidHeights_dat
    if isinstance(dat, bytes):
        dat = _BytesIO(dat)

    try:
        dat.seek(0, _SEEK_SET)  # reset
    except AttributeError as x:
        raise GeoidError(GeoidHeights_dat=type(dat), txt=str(x))

    for t in dat.readlines():
        t = t.strip()
        if t and not t.startswith(b'#'):
            lat, lon, egm84, egm96, egm2008 = map(float, t.split())
            while lon > 180:  # EasternLon to earth lon
                lon -= 360
            yield GeoidHeight5Tuple(lat, lon, egm84, egm96, egm2008)


__all__ += _ALL_DOCS(GeoidHeight5Tuple, _GeoidBase)

if __name__ == '__main__':

    import sys

    _crop     = ()
    _GeoidEGM = GeoidKarney
    _kind     = 3

    geoids = sys.argv[1:]
    while geoids:
        geoid = geoids.pop(0)

        if '-crop'.startswith(geoid.lower()):
            _crop = 20, -125, 50, -65  # CONUS

        elif '-karney'.startswith(geoid.lower()):
            _GeoidEGM = GeoidKarney

        elif '-kind'.startswith(geoid.lower()):
            _kind = int(geoids.pop(0))

        elif '-pgm'.startswith(geoid.lower()):
            _GeoidEGM = GeoidPGM

        elif geoid[-4:].lower() in ('.pgm',):
            g = _GeoidEGM(geoid, crop=_crop, kind=_kind)
            print('\n%s\n' % (g.toStr(),))
            print('%r\n' % (g.pgm,))
            # <https://GeographicLib.SourceForge.io/cgi-bin/GeoidEval>:
            # The height of the EGM96 geoid at Timbuktu
            #    echo 16:46:33N 3:00:34W | GeoidEval
            #    => 28.7068 -0.02e-6 -1.73e-6
            # The 1st number is the height of the geoid, the 2nd and
            # 3rd are its slopes in northerly and easterly direction
            t = 'Timbuktu %s' % (g,)
            k = {'egm84-15.pgm':  '31.2979',
                 'egm96-5.pgm':   '28.7067',
                 'egm2008-1.pgm': '28.7880'}.get(g.name.lower(), '28.7880')
            ll = parseDMS2('16:46:33N', '3:00:34W', sep=':')
            for ll in (ll, (16.776, -3.009),):
                try:
                    h, ll = g.height(*ll), fstr(ll, prec=6)
                    print('%s.height(%s): %.4F vs %s' % (t, ll, h, k))
                except (GeoidError, RangeError) as x:
                    print(_item_cs(t, str(x)))

        elif geoid[-4:].lower() in ('.bin',):
            g = GeoidG2012B(geoid, kind=_kind)
            print(g.toStr())

        else:
            raise GeoidError('unknown grid', txt=repr(geoid))

_I = int    # PYCHOK unused _I
del _intCs  # trash ints cache

# **) MIT License
#
# Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved.
#
# Permission is hereby granted, free of charge, to any person obtaining a
# copy of this software and associated documentation files (the "Software"),
# to deal in the Software without restriction, including without limitation
# the rights to use, copy, modify, merge, publish, distribute, sublicense,
# and/or sell copies of the Software, and to permit persons to whom the
# Software is furnished to do so, subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included
# in all copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
# OTHER DEALINGS IN THE SOFTWARE.

# <https://GeographicLib.SourceForge.io/cgi-bin/GeoidEval>
# _lowerleft = -90, -179, -30.1500  # egm2008-1.pgm
# _lowerleft = -90, -179, -29.5350  # egm96-5.pgm
# _lowerleft = -90, -179, -29.7120  # egm84-15.pgm

# _center = 0, 0, 17.2260  # egm2008-1.pgm
# _center = 0, 0, 17.1630  # egm96-5.pgm
# _center = 0, 0, 18.3296  # egm84-15.pgm

# _upperright = 90, 180, 14.8980  # egm2008-1.pgm
# _upperright = 90, 180, 13.6050  # egm96-5.pgm
# _upperright = 90, 180, 13.0980  # egm84-15.pgm

# % python pygeodesy/geoids.py [-Karney] ../geoids/egm*.pgm
#
# GeoidKarney('egm2008-1.pgm'): lowerleft(-90.0, -180.0, -30.15), upperright(90.0, 180.0, 14.898), center(0.0, 0.0, 17.226),
#                               highest(-8.4, -32.633, 85.839), lowest(4.683, -101.25, -106.911), mean=-1.317, stdev=29.244,
#                               kind=3, dtype='ushort', endian='>H', hits=0, knots=233301600, sizeB=466603604
# _PGM('../geoids/egm2008-1.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-31 06:54:00', Description='WGS84 EGM2008, 1-minute grid',
#                                  Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.025, MaxCubicError=0.003,
#                                  Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.001, RMSCubicError=0.001, Scale=0.003,
#                                  URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm2008', Vertical_Datum='WGS84', crop4=(-90.0, -180.0, 90.0, 180.0),
#                                  dlat=-0.016666666666666666, dlon=0.016666666666666666, egm=None, flon=0, glon=180, knots=233301600, nlat=10801, nlon=21600,
#                                  pgm='../geoids/egm2008-1.pgm', rlat=-60.0, rlon=60.0, sizeB=466603604, skip=404, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.775833, -3.009444): 28.7881 vs 28.7880
# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.776, -3.009): 28.7880 vs 28.7880
#
# GeoidKarney('egm84-15.pgm'): lowerleft(-90.0, -180.0, -29.712), upperright(90.0, 180.0, 13.098), center(0.0, 0.0, 18.33),
#                              highest(-8.4, -32.633, 85.839), lowest(4.683, -101.25, -106.911), mean=-1.317, stdev=29.244,
#                              kind=3, dtype='ushort', endian='>H', hits=0, knots=1038240, sizeB=2076896
# _PGM('../geoids/egm84-15.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-29 18:45:02', Description='WGS84 EGM84, 15-minute grid',
#                                 Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.413, MaxCubicError=0.02,
#                                 Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.018, RMSCubicError=0.001, Scale=0.003,
#                                 URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html', Vertical_Datum='WGS84', crop4=(-90.0, -180.0, 90.0, 180.0),
#                                 dlat=-0.25, dlon=0.25, egm=None, flon=0, glon=180, knots=1038240, nlat=721, nlon=1440,
#                                 pgm='../geoids/egm84-15.pgm', rlat=-4.0, rlon=4.0, sizeB=2076896, skip=416, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidKarney('egm84-15.pgm').height(16.775833, -3.009444): 31.2983 vs 31.2979
# Timbuktu GeoidKarney('egm84-15.pgm').height(16.776, -3.009): 31.2979 vs 31.2979
#
# GeoidKarney('egm96-5.pgm'): lowerleft(-90.0, -180.0, -29.535), upperright(90.0, 180.0, 13.605), center(0.0, 0.0, 17.163),
#                             highest(-8.4, -32.633, 85.839), lowest(4.683, -101.25, -106.911), mean=-1.317, stdev=29.244,
#                             kind=3, dtype='ushort', endian='>H', hits=0, knots=9335520, sizeB=18671448
# _PGM('../geoids/egm96-5.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-29 18:45:03', Description='WGS84 EGM96, 5-minute grid',
#                                Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.14, MaxCubicError=0.003,
#                                Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.005, RMSCubicError=0.001, Scale=0.003,
#                                URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm96/egm96.html', Vertical_Datum='WGS84', crop4=(),
#                                dlat=-0.08333333333333333, dlon=0.08333333333333333, egm=None, flon=0, glon=180, knots=9335520, nlat=2161, nlon=4320,
#                                pgm='../geoids/egm96-5.pgm', rlat=-12.0, rlon=12.0, sizeB=18671448, skip=408, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidKarney('egm96-5.pgm').height(16.775833, -3.009444): 28.7068 vs 28.7067
# Timbuktu GeoidKarney('egm96-5.pgm').height(16.776, -3.009): 28.7067 vs 28.7067

# % python pygeodesy/geoids.py -PGM ../geoids/egm*.pgm
#
# GeoidPGM('egm2008-1.pgm'): lowerleft(-90.0, -180.0, -30.15), upperright(90.0, 180.0, 14.898), center(0.0, 0.0, 17.226),
#                            highest(-8.4, -32.633, 85.839), lowest(4.683, -101.25, -106.911), mean=-1.317, stdev=29.244,
#                            kind=3, smooth=0, dtype=dtype('float64'), endian='>u2', knots=233301600, nBytes=1866412800, sizeB=466603604, scipy='1.2.1', numpy='1.16.1'
# _PGM('../geoids/egm2008-1.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-31 06:54:00', Description='WGS84 EGM2008, 1-minute grid',
#                                  Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.025, MaxCubicError=0.003,
#                                  Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.001, RMSCubicError=0.001, Scale=0.003,
#                                  URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm2008', Vertical_Datum='WGS84', crop4=(-90.0, -180.0, 90.0, 180.0),
#                                  dlat=-0.016666666666666666, dlon=0.016666666666666666, egm=None, flon=0, glon=180, knots=233301600, nlat=10801, nlon=21600,
#                                  pgm='../geoids/egm2008-1.pgm', rlat=-60.0, rlon=60.0, sizeB=466603604, skip=404, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidPGM('egm2008-1.pgm').height(16.775833, -3.009444): 28.7881 vs 28.7880
# Timbuktu GeoidPGM('egm2008-1.pgm').height(16.776, -3.009): 28.7880 vs 28.7880
#
# GeoidPGM('egm84-15.pgm'): lowerleft(-90.0, -180.0, -29.712), upperright(90.0, 180.0, 13.098), center(0.0, 0.0, 18.33),
#                           highest(-4.5, -31.25, 81.33), lowest(4.75, -100.75, -107.34), mean=-0.855, stdev=29.183,
#                           kind=3, smooth=0, dtype=dtype('float64'), endian='>u2', knots=1038240, nBytes=8305920, sizeB=2076896, scipy='1.2.1', numpy='1.16.1'
# _PGM('../geoids/egm84-15.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-29 18:45:02', Description='WGS84 EGM84, 15-minute grid',
#                                 Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.413, MaxCubicError=0.02,
#                                 Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.018, RMSCubicError=0.001, Scale=0.003,
#                                 URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html', Vertical_Datum='WGS84', crop4=(-90.0, -180.0, 90.0, 180.0),
#                                 dlat=-0.25, dlon=0.25, egm=None, flon=0, glon=180, knots=1038240, nlat=721, nlon=1440,
#                                 pgm='../geoids/egm84-15.pgm', rlat=-4.0, rlon=4.0, sizeB=2076896, skip=416, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidPGM('egm84-15.pgm').height(16.775833, -3.009444): 31.2979 vs 31.2979
# Timbuktu GeoidPGM('egm84-15.pgm').height(16.776, -3.009): 31.2975 vs 31.2979
#
# GeoidPGM('egm96-5.pgm'): lowerleft(-90.0, -180.0, -29.535), upperright(90.0, 180.0, 13.605), center(0.0, -0.0, 17.179),
#                          highest(-8.167, -32.75, 85.422), lowest(4.667, -101.167, -107.043), mean=-1.438, stdev=29.227,
#                          kind=3, smooth=0, dtype=dtype('float64'), endian='>u2', knots=9335520, nBytes=74684160, sizeB=18671448, scipy='1.2.1', numpy='1.16.1'
# _PGM('../geoids/egm96-5.pgm'): AREA_OR_POINT='Point', DateTime='2009-08-29 18:45:03', Description='WGS84 EGM96, 5-minute grid',
#                                Geoid='file in PGM format for the GeographicLib::Geoid class', MaxBilinearError=0.14, MaxCubicError=0.003,
#                                Offset=-108.0, Origin=(90, 0.0), Pixel=65535, RMSBilinearError=0.005, RMSCubicError=0.001, Scale=0.003,
#                                URL='https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm96/egm96.html', Vertical_Datum='WGS84', crop4=(-90.0, -180.0, 90.0, 180.0),
#                                dlat=-0.08333333333333333, dlon=0.08333333333333333, egm=None, flon=0, glon=180, knots=9335520, nlat=2161, nlon=4320,
#                                pgm='../geoids/egm96-5.pgm', rlat=-12.0, rlon=12.0, sizeB=18671448, skip=408, slat=90, u2B=2, wlon=0.0
# Timbuktu GeoidPGM('egm96-5.pgm').height(16.775833, -3.009444): 28.7065 vs 28.7067
# Timbuktu GeoidPGM('egm96-5.pgm').height(16.776, -3.009): 28.7064 vs 28.7067