Coverage for pygeodesy/geoids.py: 96%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Geoid models and geoid height interpolations. 

5 

6Classes L{GeoidG2012B}, L{GeoidKarney} and L{GeoidPGM} to interpolate the 

7height of various U{geoid<https://WikiPedia.org/wiki/Geoid>}s at C{LatLon} 

8locations or separate lat-/longitudes using different interpolation methods 

9and C{geoid} model files. 

10 

11L{GeoidKarney} is a transcoding of I{Charles Karney}'s C++ class U{Geoid 

12<https://GeographicLib.SourceForge.io/C++/doc/geoid.html>} to pure Python. 

13The L{GeoidG2012B} and L{GeoidPGM} interpolators both depend on U{scipy 

14<https://SciPy.org>} and U{numpy<https://PyPI.org/project/numpy>} and 

15require those packages to be installed. 

16 

17In addition, each geoid interpolator needs C{grid knots} (down)loaded from 

18a C{geoid} model file, I{specific to the interpolator}, more details below 

19and in the documentation of the interpolator class. For each interpolator, 

20there are several interpolation choices, like I{linear}, I{cubic}, etc. 

21 

22Typical usage 

23============= 

24 

251. Choose one of the interpolator classes L{GeoidG2012B}, L{GeoidKarney} 

26or L{GeoidPGM} and download a C{geoid} model file, containing locations with 

27known heights also referred to as the C{grid knots}. See the documentation 

28of the interpolator class for references to available C{grid} models. 

29 

30C{>>> from pygeodesy import GeoidG2012B # or -Karney or -PGM as GeoidXyz} 

31 

322. Instantiate an interpolator with the C{geoid} model file and use keyword 

33arguments to select different interpolation options 

34 

35C{>>> ginterpolator = GeoidXyz(geoid_model_file, **options)} 

36 

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

38 

39C{>>> ll = LatLon(1, 2, ...)} 

40 

41C{>>> h = ginterpolator(ll)} 

42 

43or 

44 

45C{>>> h0, h1, h2, ... = ginterpolator(ll0, ll1, ll2, ...)} 

46 

47or a list, tuple, generator, etc. of C{LatLon}s 

48 

49C{>>> hs = ginterpolator(lls)} 

50 

514. For separate lat- and longitudes invoke the C{.height} method as 

52 

53C{>>> h = ginterpolator.height(lat, lon)} 

54 

55or as 2 lists, 2 tuples, etc. 

56 

57C{>>> hs = ginterpolator.height(lats, lons)} 

58 

595. An example is in U{issue #64<https://GitHub.com/mrJean1/PyGeodesy/issues/64>}, 

60courtesy of SBFRF. 

61 

62@note: Classes L{GeoidG2012B} and L{GeoidPGM} require both U{numpy 

63 <https://PyPI.org/project/numpy>} and U{scipy<https://PyPI.org/project/scipy>} 

64 to be installed. 

65 

66@note: Errors from C{scipy} are raised as L{SciPyError}s. Warnings issued by 

67 C{scipy} can be thrown as L{SciPyWarning} exceptions, provided Python 

68 C{warnings} are filtered accordingly, see L{SciPyWarning}. 

69 

70@see: I{Karney}'s U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}, 

71 U{Geoid height<https://GeographicLib.SourceForge.io/C++/doc/geoid.html>} and U{Installing 

72 the Geoid datasets<https://GeographicLib.SourceForge.io/C++/doc/geoid.html#geoidinst>}, 

73 U{SciPy<https://docs.SciPy.org/doc/scipy/reference/interpolate.html>} interpolation 

74 U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/reference/generated/scipy. 

75 interpolate.RectBivariateSpline.html>} and U{interp2d<https://docs.SciPy.org/doc/scipy/ 

76 reference/generated/scipy.interpolate.interp2d.html>}, functions L{elevations.elevation2} 

77 and L{elevations.geoidHeight2}, U{I{Ellispoid vs Orthometric Elevations}<https:// 

78 www.YouTube.com/watch?v=dX6a6kCk3Po>} and U{I{Pitfalls Related to Ellipsoid Height 

79 and Height Above Mean Sea Level (AMSL)}<https://Wiki.ROS.org/mavros#mavros.2FPlugins. 

80 Avoiding_Pitfalls_Related_to_Ellipsoid_Height_and_Height_Above_Mean_Sea_Level>}. 

81''' 

82# make sure int/int division yields float quotient, see .basics 

83from __future__ import division as _; del _ # PYCHOK semicolon 

84 

85from pygeodesy.basics import len2, map1, isodd, ub2str as _ub2str 

86from pygeodesy.constants import EPS, _float as _F, _0_0, _1_0, _180_0, _360_0 

87# from pygeodesy.datums import _ellipsoidal_datum # from .heights 

88# from pygeodesy.dms import parseDMS2 # _MODS 

89from pygeodesy.errors import _incompatible, LenError, RangeError, SciPyError, \ 

90 _SciPyIssue, _xkwds_pop2 

91from pygeodesy.fmath import favg, Fdot, fdot, Fhorner, frange 

92# from pygoedesy.formy import heightOrthometric # _MODS 

93from pygeodesy.heights import _as_llis2, _ascalar, _height_called, HeightError, \ 

94 _HeightsBase, _ellipsoidal_datum, _Wrap 

95# from pygeodesy.internals import _version2 # _MODS 

96from pygeodesy.interns import MISSING, NN, _4_, _COLONSPACE_, _COMMASPACE_, \ 

97 _cubic_, _E_, _height_, _in_, _kind_, _lat_, \ 

98 _linear_, _lon_, _mean_, _N_, _n_a_, _numpy_, \ 

99 _on_, _outside_, _S_, _s_, _scipy_, _SPACE_, \ 

100 _stdev_, _tbd_, _W_, _width_ 

101from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS, _FOR_DOCS 

102from pygeodesy.named import _name__, _Named, _NamedTuple 

103# from pygeodesy.namedTuples import LatLon3Tuple # _MODS 

104from pygeodesy.props import deprecated_method, Property_RO, property_RO 

105from pygeodesy.streprs import attrs, Fmt, fstr, pairs 

106from pygeodesy.units import Height, Int_, Lat, Lon 

107# from pygeodesy.utily import _Wrap # from .heights 

108 

109from math import floor 

110import os.path as _os_path 

111from os import SEEK_CUR as _SEEK_CUR, SEEK_SET as _SEEK_SET 

112from struct import calcsize as _calcsize, unpack as _unpack 

113try: 

114 from StringIO import StringIO as _BytesIO # reads bytes 

115 _ub2str = str # PYCHOK convert bytes to str for egm*.pgm text 

116 

117except ImportError: # Python 3+ 

118 from io import BytesIO as _BytesIO # PYCHOK expected 

119 

120__all__ = _ALL_LAZY.geoids 

121__version__ = '24.06.03' 

122 

123_assert_ = 'assert' 

124_bHASH_ = b'#' 

125_endian_ = 'endian' 

126_format_ = '%s %r' 

127_header_ = 'header' 

128_intCs = {} # cache int value 

129_interp2d_ks = {-2: _linear_, 

130 -3: _cubic_, 

131 -5: 'quintic'} 

132_lli_ = 'lli' 

133_non_increasing_ = 'non-increasing' 

134_rb_ = 'rb' 

135_supported_ = 'supported' 

136 

137 

138class _GeoidBase(_HeightsBase): 

139 '''(INTERNAL) Base class for C{Geoid...}s. 

140 ''' 

141 _cropped = None 

142# _datum = _WGS84 # from _HeightsBase 

143 _egm = None # open C{egm*.pgm} geoid file 

144 _endian = _tbd_ 

145 _geoid = _n_a_ 

146 _hs_y_x = None # numpy 2darray, row-major order 

147 _interp2d = None # interp2d interpolation 

148 _kind = 3 # order for interp2d, RectBivariateSpline 

149# _kmin = 2 # min number of knots 

150 _knots = 0 # nlat * nlon 

151 _mean = None # fixed in GeoidKarney 

152# _name = NN # _Named 

153 _nBytes = 0 # numpy size in bytes, float64 

154 _pgm = None # PGM attributes, C{_PGM} or C{None} 

155 _sizeB = 0 # geoid file size in bytes 

156 _smooth = 0 # used only for RectBivariateSpline 

157 _stdev = None # fixed in GeoidKarney 

158 _u2B = 0 # np.itemsize or undefined 

159 

160 _lat_d = _0_0 # increment, +tive 

161 _lat_lo = _0_0 # lower lat, south 

162 _lat_hi = _0_0 # upper lat, noth 

163 _lon_d = _0_0 # increment, +tive 

164 _lon_lo = _0_0 # left lon, west 

165 _lon_hi = _0_0 # right lon, east 

166 _lon_of = _0_0 # forward lon offset 

167 _lon_og = _0_0 # reverse lon offset 

168 

169 _center = None # (lat, lon, height) 

170 _yx_hits = None # cache hits, ala Karney 

171 

172 def __init__(self, hs, p): 

173 '''(INTERNAL) Set up the grid axes, the C{SciPy} interpolator 

174 and several internal geoid attributes. 

175 

176 @arg hs: Grid knots with known height (C{numpy 2darray}). 

177 @arg p: The C{slat, wlon, nlat, nlon, dlat, dlon} and 

178 other geoid parameters (C{INTERNAL}). 

179 

180 @raise GeoidError: Incompatible grid B{C{hs}} shape or 

181 invalid B{C{kind}}. 

182 

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

184 

185 @raise SciPyError: A C{scipy.interpolate.inter2d} or 

186 C{-.RectBivariateSpline} issue. 

187 

188 @raise SciPyWarning: A C{scipy.interpolate.inter2d} or 

189 C{-.RectBivariateSpline} warning as 

190 exception. 

191 

192 @note: C{scipy.interpolate.interp2d} has been C{DEPRECATED}, 

193 specify keyword argument C{B{kind}=1..5} to use 

194 C{scipy.interpolate.RectBivariateSpline}. 

195 ''' 

196 spi = self.scipy_interpolate 

197 # for 2d scipy.interpolate.interp2d(xs, ys, hs, ...) and 

198 # scipy.interpolate.RectBivariateSpline(ys, xs, hs, ...) 

199 # require the shape of hs to be (len(ys), len(xs)), note 

200 # the different (xs, ys, ...) and (ys, xs, ...) orders 

201 if (p.nlat, p.nlon) != hs.shape: 

202 raise GeoidError(shape=hs.shape, txt=_incompatible((p.nlat, p.nlon))) 

203 

204 # both axes and bounding box 

205 ys, self._lat_d = self._gaxis2(p.slat, p.dlat, p.nlat, _lat_ + _s_) 

206 xs, self._lon_d = self._gaxis2(p.wlon, p.dlon, p.nlon, _lon_ + _s_) 

207 

208 bb = ys[0], ys[-1], xs[0], xs[-1] + p.dlon # fudge lon_hi 

209 # geoid grids are typically stored in row-major order, some 

210 # with rows (90..-90) reversed and columns (0..360) wrapped 

211 # to Easten longitude, 0 <= east < 180 and 180 <= west < 360 

212 k = self.kind 

213 if k in _interp2d_ks: # .interp2d DEPRECATED since scipy 1.10 

214 if self._scipy_version() < (1, 10): 

215 self._interp2d = spi.interp2d(xs, ys, hs, kind=_interp2d_ks[k]) 

216 else: # call and overwrite the DEPRECATED .interp2d 

217 self._interp2d = self._interp2d(xs, ys, hs, k) 

218 elif 1 <= k <= 5: 

219 self._ev = spi.RectBivariateSpline(ys, xs, hs, bbox=bb, ky=k, kx=k, 

220 s=self._smooth).ev 

221 else: 

222 raise GeoidError(kind=k) 

223 

224 self._hs_y_x = hs # numpy 2darray, row-major 

225 self._nBytes = hs.nbytes # numpy size in bytes 

226 self._knots = p.knots # grid knots 

227 self._lon_of = float(p.flon) # forward offset 

228 self._lon_og = float(p.glon) # reverse offset 

229 # shrink the box by 1 unit on every side 

230 # bb += self._lat_d, -self._lat_d, self._lon_d, -self._lon_d 

231 self._lat_lo = float(bb[0]) 

232 self._lat_hi = float(bb[1]) 

233 self._lon_lo = float(bb[2] - p.glon) 

234 self._lon_hi = float(bb[3] - p.glon) 

235 

236 def __call__(self, *llis, **wrap_H): 

237 '''Interpolate the geoid height for one or several locations. 

238 

239 @arg llis: One or more locations (C{LatLon}s), all positional. 

240 @kwarg wrap_H: Keyword arguments C{B{wrap}=False} (C{bool}) and 

241 C{B{H}=False} (C{bool}). If C{B{wrap} is True}, 

242 wrap or I{normalize} all B{C{llis}} locations. If 

243 C{B{H} is True}, return the I{orthometric} height 

244 instead of the I{geoid} height at each location. 

245 

246 @return: A single interpolated geoid (or orthometric) height 

247 (C{float}) or a list or tuple of interpolated geoid 

248 (or orthometric) heights (C{float}s). 

249 

250 @raise GeoidError: Insufficient number of B{C{llis}}, an 

251 invalid B{C{lli}} or the C{egm*.pgm} 

252 geoid file is closed. 

253 

254 @raise RangeError: An B{C{lli}} is outside this geoid's lat- 

255 or longitude range. 

256 

257 @raise SciPyError: A C{scipy.interpolate.inter2d} or 

258 C{-.RectBivariateSpline} issue. 

259 

260 @raise SciPyWarning: A C{scipy.interpolate.inter2d} or 

261 C{-.RectBivariateSpline} warning as 

262 exception. 

263 

264 @note: To obtain I{orthometric} heights, each B{C{llis}} 

265 location must have an ellipsoid C{height} or C{h} 

266 attribute, otherwise C{height=0} is used. 

267 

268 @see: Function L{pygeodesy.heightOrthometric}. 

269 ''' 

270 return self._called(llis, True, **wrap_H) 

271 

272 def __enter__(self): 

273 '''Open context. 

274 ''' 

275 return self 

276 

277 def __exit__(self, *unused): # PYCHOK exc_type, exc_value, exc_traceback) 

278 '''Close context. 

279 ''' 

280 self.close() 

281 # return None # XXX False 

282 

283 def __repr__(self): 

284 return self.toStr() 

285 

286 def __str__(self): 

287 return Fmt.PAREN(self.classname, repr(self.name)) 

288 

289 def _called(self, llis, scipy, wrap=False, H=False): 

290 # handle __call__ 

291 _H = _MODS.formy.heightOrthometric if H else None 

292 _as, llis = _as_llis2(llis, Error=GeoidError) 

293 hs, _w = [], _Wrap._latlonop(wrap) 

294 _a, _h = hs.append, self._hGeoid 

295 try: 

296 for i, lli in enumerate(llis): 

297 N = _h(*_w(lli.lat, lli.lon)) 

298 # orthometric or geoid height 

299 _a(_H(lli, N) if _H else N) 

300 return _as(hs) 

301 

302 except (GeoidError, RangeError) as x: 

303 # XXX avoid str(LatLon()) degree symbols 

304 t = _lli_ if _as is _ascalar else Fmt.INDEX(llis=i) 

305 lli = fstr((lli.lat, lli.lon), strepr=repr) 

306 raise type(x)(t, lli, wrap=wrap, H=H, cause=x) 

307 except Exception as x: 

308 if scipy and self.scipy: 

309 raise _SciPyIssue(x) 

310 else: 

311 raise 

312 

313 @Property_RO 

314 def _center(self): 

315 ''' Cache for method L{center}. 

316 ''' 

317 return self._llh3(favg(self._lat_lo, self._lat_hi), 

318 favg(self._lon_lo, self._lon_hi)) 

319 

320 def center(self, LatLon=None): 

321 '''Return the center location and height of this geoid. 

322 

323 @kwarg LatLon: Optional class to return the location and height 

324 (C{LatLon}) or C{None}. 

325 

326 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

327 lon, height)} otherwise a B{C{LatLon}} instance 

328 with the lat-, longitude and geoid height of the 

329 center grid location. 

330 ''' 

331 return self._llh3LL(self._center, LatLon) 

332 

333 def close(self): 

334 '''Close the C{egm*.pgm} geoid file if open (and applicable). 

335 ''' 

336 if not self.closed: 

337 self._egm.close() 

338 self._egm = None 

339 

340 @property_RO 

341 def closed(self): 

342 '''Get the C{egm*.pgm} geoid file status. 

343 ''' 

344 return self._egm is None 

345 

346 @Property_RO 

347 def cropped(self): 

348 '''Is geoid cropped (C{bool} or C{None} if crop not supported). 

349 ''' 

350 return self._cropped 

351 

352 @Property_RO 

353 def dtype(self): 

354 '''Get the grid C{scipy} U{dtype<https://docs.SciPy.org/doc/numpy/ 

355 reference/generated/numpy.ndarray.dtype.html>} (C{numpy.dtype}). 

356 ''' 

357 return self._hs_y_x.dtype 

358 

359 @Property_RO 

360 def endian(self): 

361 '''Get the geoid endianess and U{dtype<https://docs.SciPy.org/ 

362 doc/numpy/reference/generated/numpy.dtype.html>} (C{str}). 

363 ''' 

364 return self._endian 

365 

366 def _ev(self, y, x): # PYCHOK expected 

367 # only used for .interpolate.interp2d, but 

368 # overwritten for .RectBivariateSpline, 

369 # note (y, x) must be flipped! 

370 return self._interp2d(x, y) 

371 

372 def _gaxis2(self, lo, d, n, name): 

373 # build grid axis, hi = lo + (n - 1) * d 

374 m, a = len2(frange(lo, n, d)) 

375 if m != n: 

376 raise LenError(self.__class__, grid=m, **{name: n}) 

377 if d < 0: 

378 d, a = -d, list(reversed(a)) 

379 for i in range(1, m): 

380 e = a[i] - a[i-1] 

381 if e < EPS: # non-increasing axis 

382 i = Fmt.INDEX(name, i) 

383 raise GeoidError(i, e, txt=_non_increasing_) 

384 return self.numpy.array(a), d 

385 

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

387 '''(INTERNAL) I{Must be overloaded}.''' 

388 self._notOverloaded(lat, lon) 

389 

390 def _gyx2g2(self, y, x): 

391 # convert grid (y, x) indices to grid (lat, lon) 

392 return ((self._lat_lo + self._lat_d * y), 

393 (self._lon_lo + self._lon_of + self._lon_d * x)) 

394 

395 def height(self, lats, lons, **wrap): 

396 '''Interpolate the geoid height for one or several lat-/longitudes. 

397 

398 @arg lats: Latitude or latitudes (C{degrees} or C{degrees}s). 

399 @arg lons: Longitude or longitudes (C{degrees} or C{degrees}s). 

400 @kwarg wrap: If C{True}, wrap or I{normalize} all B{C{lats}} 

401 and B{C{lons}} locations (C{bool}). 

402 

403 @return: A single interpolated geoid height (C{float}) or a 

404 list of interpolated geoid heights (C{float}s). 

405 

406 @raise GeoidError: Insufficient or non-matching number of 

407 B{C{lats}} and B{C{lons}}. 

408 

409 @raise RangeError: A B{C{lat}} or B{C{lon}} is outside this 

410 geoid's lat- or longitude range. 

411 

412 @raise SciPyError: A C{scipy.interpolate.inter2d} or 

413 C{-.RectBivariateSpline} issue. 

414 

415 @raise SciPyWarning: A C{scipy.interpolate.inter2d} or 

416 C{-.RectBivariateSpline} warning as 

417 exception. 

418 ''' 

419 return _height_called(self, lats, lons, Error=GeoidError, **wrap) 

420 

421 def _hGeoid(self, lat, lon): 

422 out = self.outside(lat, lon) 

423 if out: 

424 lli = fstr((lat, lon), strepr=repr) 

425 raise RangeError(lli=lli, txt=_SPACE_(_outside_, _on_, out)) 

426 return float(self._ev(*self._ll2g2(lat, lon))) 

427 

428 @Property_RO 

429 def _highest(self): 

430 '''(INTERNAL) Cache for L{highest} method. 

431 ''' 

432 return self._llh3minmax(True) 

433 

434 def highest(self, LatLon=None, **unused): 

435 '''Return the location and largest height of this geoid. 

436 

437 @kwarg LatLon: Optional class to return the location and height 

438 (C{LatLon}) or C{None}. 

439 

440 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

441 lon, height)} otherwise a B{C{LatLon}} instance 

442 with the lat-, longitude and geoid height of the 

443 highest grid location. 

444 ''' 

445 return self._llh3LL(self._highest, LatLon) 

446 

447 @Property_RO 

448 def hits(self): 

449 '''Get the number of cache hits (C{int} or C{None}). 

450 ''' 

451 return self._yx_hits 

452 

453 @deprecated_method 

454 def _interp2d(self, xs, ys, hs=(), k=0): # overwritten in .__init__ above 

455 '''DEPRECATED on 23.01.06, use keyword argument C{B{kind}=1..5}.''' 

456 # assert k in _interp2d_ks # and len(hs) == len(xs) == len(ys) 

457 try: 

458 return self.scipy_interpolate.interp2d(xs, ys, hs, kind=_interp2d_ks[k]) 

459 except AttributeError as x: 

460 raise SciPyError(interp2d=MISSING, kind=k, cause=x) 

461 

462 @Property_RO 

463 def kind(self): 

464 '''Get the interpolator kind and order (C{int}). 

465 ''' 

466 return self._kind 

467 

468 @Property_RO 

469 def knots(self): 

470 '''Get the number of grid knots (C{int}). 

471 ''' 

472 return self._knots 

473 

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

475 '''(INTERNAL) I{Must be overloaded}.''' 

476 self._notOverloaded(lat, lon) 

477 

478 @property_RO 

479 def _LL3T(self): 

480 '''(INTERNAL) Get L{LatLon3Tuple}, I{once}. 

481 ''' 

482 _GeoidBase._LL3T = T = _MODS.namedTuples.LatLon3Tuple # overwrite poperty_RO 

483 return T 

484 

485 def _llh3(self, lat, lon): 

486 return self._LL3T(lat, lon, self._hGeoid(lat, lon), name=self.name) 

487 

488 def _llh3LL(self, llh, LatLon): 

489 return llh if LatLon is None else self._xnamed(LatLon(*llh)) 

490 

491 def _llh3minmax(self, highest=True, *unused): 

492 hs, np = self._hs_y_x, self.numpy 

493 # <https://docs.SciPy.org/doc/numpy/reference/generated/ 

494 # numpy.argmin.html#numpy.argmin> 

495 arg = np.argmax if highest else np.argmin 

496 y, x = np.unravel_index(arg(hs, axis=None), hs.shape) 

497 return self._g2ll2(*self._gyx2g2(y, x)) + (float(hs[y, x]),) 

498 

499 def _load(self, g, dtype, n, offset=0): 

500 # numpy.fromfile, like .frombuffer 

501 g.seek(offset, _SEEK_SET) 

502 return self.numpy.fromfile(g, dtype, n) 

503 

504 @Property_RO 

505 def _lowerleft(self): 

506 '''(INTERNAL) Cache for L{lowerleft}. 

507 ''' 

508 return self._llh3(self._lat_lo, self._lon_lo) 

509 

510 def lowerleft(self, LatLon=None): 

511 '''Return the lower-left location and height of this geoid. 

512 

513 @kwarg LatLon: Optional class to return the location 

514 (C{LatLon}) and height or C{None}. 

515 

516 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

517 lon, height)} otherwise a B{C{LatLon}} instance 

518 with the lat-, longitude and geoid height of the 

519 lower-left, SW grid corner. 

520 ''' 

521 return self._llh3LL(self._lowerleft, LatLon) 

522 

523 @Property_RO 

524 def _loweright(self): 

525 '''(INTERNAL) Cache for L{loweright}. 

526 ''' 

527 return self._llh3(self._lat_lo, self._lon_hi) 

528 

529 def loweright(self, LatLon=None): 

530 '''Return the lower-right location and height of this geoid. 

531 

532 @kwarg LatLon: Optional class to return the location and height 

533 (C{LatLon}) or C{None}. 

534 

535 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

536 lon, height)} otherwise a B{C{LatLon}} instance 

537 with the lat-, longitude and geoid height of the 

538 lower-right, SE grid corner. 

539 ''' 

540 

541 return self._llh3LL(self._loweright, LatLon) 

542 

543 lowerright = loweright # synonymous 

544 

545 @Property_RO 

546 def _lowest(self): 

547 '''(INTERNAL) Cache for L{lowest}. 

548 ''' 

549 return self._llh3minmax(False) 

550 

551 def lowest(self, LatLon=None, **unused): 

552 '''Return the location and lowest height of this geoid. 

553 

554 @kwarg LatLon: Optional class to return the location and height 

555 (C{LatLon}) or C{None}. 

556 

557 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

558 lon, height)} otherwise a B{C{LatLon}} instance 

559 with the lat-, longitude and geoid height of the 

560 lowest grid location. 

561 ''' 

562 return self._llh3LL(self._lowest, LatLon) 

563 

564 @Property_RO 

565 def mean(self): 

566 '''Get the mean of this geoid's heights (C{float}). 

567 ''' 

568 if self._mean is None: # see GeoidKarney 

569 self._mean = float(self.numpy.mean(self._hs_y_x)) 

570 return self._mean 

571 

572 @property_RO 

573 def name(self): 

574 '''Get the name of this geoid (C{str}). 

575 ''' 

576 return _HeightsBase.name.fget(self) or self._geoid # recursion 

577 

578 @Property_RO 

579 def nBytes(self): 

580 '''Get the grid in-memory size in bytes (C{int}). 

581 ''' 

582 return self._nBytes 

583 

584 def _open(self, geoid, datum, kind, name, smooth): 

585 # open the geoid file 

586 try: 

587 self._geoid = _os_path.basename(geoid) 

588 self._sizeB = _os_path.getsize(geoid) 

589 g = open(geoid, _rb_) 

590 except (IOError, OSError) as x: 

591 raise GeoidError(geoid=geoid, cause=x) 

592 

593 if datum not in (None, self._datum): 

594 self._datum = _ellipsoidal_datum(datum, name=name) 

595 self._kind = int(kind) 

596 if name: 

597 _HeightsBase.name.fset(self, name) # rename 

598 if smooth: 

599 self._smooth = Int_(smooth=smooth, Error=GeoidError, low=0) 

600 

601 return g 

602 

603 def outside(self, lat, lon): 

604 '''Check whether a location is outside this geoid's 

605 lat-/longitude or crop range. 

606 

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

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

609 

610 @return: A 1- or 2-character C{str} if outside or an 

611 empty C{str} if inside. 

612 ''' 

613 return (_S_ if lat < self._lat_lo else 

614 (_N_ if lat > self._lat_hi else NN)) + \ 

615 (_W_ if lon < self._lon_lo else 

616 (_E_ if lon > self._lon_hi else NN)) 

617 

618 @Property_RO 

619 def pgm(self): 

620 '''Get the PGM attributes (C{_PGM} or C{None} if not available/applicable). 

621 ''' 

622 return self._pgm 

623 

624 @Property_RO 

625 def sizeB(self): 

626 '''Get the geoid grid file size in bytes (C{int}). 

627 ''' 

628 return self._sizeB 

629 

630 @Property_RO 

631 def smooth(self): 

632 '''Get the C{RectBivariateSpline} smoothing (C{int}). 

633 ''' 

634 return self._smooth 

635 

636 @Property_RO 

637 def stdev(self): 

638 '''Get the standard deviation of this geoid's heights (C{float}) or C{None}. 

639 ''' 

640 if self._stdev is None: # see GeoidKarney 

641 self._stdev = float(self.numpy.std(self._hs_y_x)) 

642 return self._stdev 

643 

644 def _swne(self, crop): 

645 # crop box to 4-tuple (s, w, n, e) 

646 try: 

647 if len(crop) == 2: 

648 try: # sw, ne LatLons 

649 swne = (crop[0].lat, crop[0].lon, 

650 crop[1].lat, crop[1].lon) 

651 except AttributeError: # (s, w), (n, e) 

652 swne = tuple(crop[0]) + tuple(crop[1]) 

653 else: # (s, w, n, e) 

654 swne = crop 

655 if len(swne) == 4: 

656 s, w, n, e = map(float, swne) 

657 if -90 <= s <= (n - _1_0) <= 89 and \ 

658 -180 <= w <= (e - _1_0) <= 179: 

659 return s, w, n, e 

660 except (IndexError, TypeError, ValueError): 

661 pass 

662 raise GeoidError(crop=crop) 

663 

664 def toStr(self, prec=3, sep=_COMMASPACE_): # PYCHOK signature 

665 '''This geoid and all geoid attributes as a string. 

666 

667 @kwarg prec: Number of decimal digits (0..9 or C{None} for 

668 default). Trailing zero decimals are stripped 

669 for B{C{prec}} values of 1 and above, but kept 

670 for negative B{C{prec}} values. 

671 @kwarg sep: Separator to join (C{str}). 

672 

673 @return: Geoid name and attributes (C{str}). 

674 ''' 

675 s = 1 if self.kind < 0 else 2 

676 t = tuple(Fmt.PAREN(m.__name__, fstr(m(), prec=prec)) for m in 

677 (self.lowerleft, self.upperright, 

678 self.center, 

679 self.highest, self.lowest)) + \ 

680 attrs( _mean_, _stdev_, prec=prec, Nones=False) + \ 

681 attrs((_kind_, 'smooth')[:s], prec=prec, Nones=False) + \ 

682 attrs( 'cropped', 'dtype', _endian_, 'hits', 'knots', 'nBytes', 

683 'sizeB', _scipy_, _numpy_, prec=prec, Nones=False) 

684 return _COLONSPACE_(self, sep.join(t)) 

685 

686 @Property_RO 

687 def u2B(self): 

688 '''Get the PGM itemsize in bytes (C{int}). 

689 ''' 

690 return self._u2B 

691 

692 @Property_RO 

693 def _upperleft(self): 

694 '''(INTERNAL) Cache for method L{upperleft}. 

695 ''' 

696 return self._llh3(self._lat_hi, self._lon_lo) 

697 

698 def upperleft(self, LatLon=None): 

699 '''Return the upper-left location and height of this geoid. 

700 

701 @kwarg LatLon: Optional class to return the location and height 

702 (C{LatLon}) or C{None}. 

703 

704 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

705 lon, height)} otherwise a B{C{LatLon}} instance 

706 with the lat-, longitude and geoid height of the 

707 upper-left, NW grid corner. 

708 ''' 

709 return self._llh3LL(self._upperleft, LatLon) 

710 

711 @Property_RO 

712 def _upperright(self): 

713 '''(INTERNAL) Cache for method L{upperright}. 

714 ''' 

715 return self._llh3(self._lat_hi, self._lon_hi) 

716 

717 def upperright(self, LatLon=None): 

718 '''Return the upper-right location and height of this geoid. 

719 

720 @kwarg LatLon: Optional class to return the location and height 

721 (C{LatLon}) or C{None}. 

722 

723 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

724 lon, height)} otherwise a B{C{LatLon}} instance 

725 with the lat-, longitude and geoid height of the 

726 upper-right, NE grid corner. 

727 ''' 

728 return self._llh3LL(self._upperright, LatLon) 

729 

730 

731class GeoidError(HeightError): 

732 '''Geoid interpolator C{Geoid...} or interpolation issue. 

733 ''' 

734 pass 

735 

736 

737class GeoidG2012B(_GeoidBase): 

738 '''Geoid height interpolator for U{GEOID12B Model 

739 <https://www.NGS.NOAA.gov/GEOID/GEOID12B/>} grids U{CONUS 

740 <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_CONUS.shtml>}, 

741 U{Alaska<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_AK.shtml>}, 

742 U{Hawaii<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_HI.shtml>}, 

743 U{Guam and Northern Mariana Islands 

744 <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_GMNI.shtml>}, 

745 U{Puerto Rico and U.S. Virgin Islands 

746 <https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_PRVI.shtml>} and 

747 U{American Samoa<https://www.NGS.NOAA.gov/GEOID/GEOID12B/GEOID12B_AS.shtml>} 

748 based on C{SciPy} U{RectBivariateSpline<https://docs.SciPy.org/doc/ 

749 scipy/reference/generated/scipy.interpolate.RectBivariateSpline.html>} 

750 or U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/ 

751 scipy.interpolate.interp2d.html>} interpolation. 

752 

753 Use any of the binary C{le} (little endian) or C{be} (big endian) 

754 C{g2012b*.bin} grid files. 

755 ''' 

756 def __init__(self, g2012b_bin, datum=None, # NAD 83 Ellipsoid 

757 kind=3, smooth=0, **name_crop): 

758 '''New L{GeoidG2012B} interpolator. 

759 

760 @arg g2012b_bin: A C{GEOID12B} grid file name (C{.bin}). 

761 @kwarg datum: Optional grid datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} 

762 or L{a_f2Tuple}), default C{WGS84}. 

763 @kwarg kind: C{scipy.interpolate} order (C{int}), use 1..5 for 

764 U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/ 

765 reference/generated/scipy.interpolate.RectBivariateSpline.html>}, 

766 -2 for U{interp2d linear<https://docs.SciPy.org/doc/scipy/ 

767 reference/generated/scipy.interpolate.interp2d.html>}, -3 

768 for C{interp2d cubic} or -5 for C{interp2d quintic}. 

769 @kwarg smooth: Smoothing factor for U{RectBivariateSpline 

770 <https://docs.SciPy.org/doc/scipy/reference/generated/ 

771 scipy.interpolate.RectBivariateSpline.html>} 

772 only (C{int}). 

773 @kwarg name_crop: Optional geoid C{B{name}=NN} (C{str}) and UNSUPPORTED 

774 keyword argument C{B{crop}}, use C{B{crop}=None} to ignore. 

775 

776 @raise GeoidError: G2012B grid file B{C{g2012b_bin}} issue or invalid 

777 B{C{crop}}, B{C{kind}} or B{C{smooth}}. 

778 

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

780 installed. 

781 

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

783 

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

785 

786 @raise SciPyWarning: A C{RectBivariateSpline} or C{inter2d} 

787 warning as exception. 

788 

789 @raise TypeError: Invalid B{C{datum}}. 

790 

791 @note: C{scipy.interpolate.interp2d} has been C{DEPRECATED}, specify 

792 C{B{kind}=1..5} for C{scipy.interpolate.RectBivariateSpline}. 

793 ''' 

794 crop, name = _xkwds_pop2(name_crop, crop=None) 

795 if crop is not None: 

796 raise GeoidError(crop=crop, txt_not_=_supported_) 

797 

798 g = self._open(g2012b_bin, datum, kind, _name__(**name), smooth) 

799 _ = self.numpy # import numpy for ._load and 

800 

801 try: 

802 p = _Gpars() 

803 n = (self.sizeB // 4) - 11 # number of f4 heights 

804 # U{numpy dtype formats are different from Python struct formats 

805 # <https://docs.SciPy.org/doc/numpy-1.15.0/reference/arrays.dtypes.html>} 

806 for en_ in ('<', '>'): 

807 # skip 4xf8, get 3xi4 

808 p.nlat, p.nlon, ien = map(int, self._load(g, en_+'i4', 3, 32)) 

809 if ien == 1: # correct endian 

810 p.knots = p.nlat * p.nlon 

811 if p.knots == n and 1 < p.nlat < n \ 

812 and 1 < p.nlon < n: 

813 self._endian = en_+'f4' 

814 break 

815 else: # couldn't validate endian 

816 raise GeoidError(_endian_) 

817 

818 # get the first 4xf8 

819 p.slat, p.wlon, p.dlat, p.dlon = map(float, self._load(g, en_+'f8', 4)) 

820 # read all f4 heights, ignoring the first 4xf8 and 3xi4 

821 hs = self._load(g, self._endian, n, 44).reshape(p.nlat, p.nlon) 

822 p.wlon -= _360_0 # western-most East longitude to earth (..., lon) 

823 _GeoidBase.__init__(self, hs, p) 

824 

825 except Exception as x: 

826 raise _SciPyIssue(x, _in_, repr(g2012b_bin)) 

827 finally: 

828 g.close() 

829 

830 def _g2ll2(self, lat, lon): 

831 # convert grid (lat, lon) to earth (lat, lon) 

832 return lat, lon 

833 

834 def _ll2g2(self, lat, lon): 

835 # convert earth (lat, lon) to grid (lat, lon) 

836 return lat, lon 

837 

838 if _FOR_DOCS: 

839 __call__ = _GeoidBase.__call__ 

840 height = _GeoidBase.height 

841 

842 

843class GeoidHeight5Tuple(_NamedTuple): # .geoids.py 

844 '''5-Tuple C{(lat, lon, egm84, egm96, egm2008)} for U{GeoidHeights.dat 

845 <https://SourceForge.net/projects/geographiclib/files/testdata/>} 

846 tests with the heights for 3 different EGM grids at C{degrees90} 

847 and C{degrees180} degrees (after converting C{lon} from original 

848 C{0 <= EasterLon <= 360}). 

849 ''' 

850 _Names_ = (_lat_, _lon_, 'egm84', 'egm96', 'egm2008') 

851 _Units_ = ( Lat, Lon, Height, Height, Height) 

852 

853 

854def _I(i): 

855 '''(INTERNAL) Cache a single C{int} constant. 

856 ''' 

857 return _intCs.setdefault(i, i) # PYCHOK undefined due to del _intCs 

858 

859 

860def _T(*cs): 

861 '''(INTERNAL) Cache a tuple of single C{int} constants. 

862 ''' 

863 return map1(_I, *cs) 

864 

865_T0s12 = (_I(0),) * 12 # PYCHOK _T(0, 0, ..., 0) 

866 

867 

868class GeoidKarney(_GeoidBase): 

869 '''Geoid height interpolator for I{Karney}'s U{GeographicLib Earth 

870 Gravitational Model (EGM)<https://GeographicLib.SourceForge.io/C++/doc/ 

871 geoid.html>} geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/ 

872 C++/doc/geoid.html#geoidinst>} datasets using bilinear or U{cubic 

873 <https://dl.ACM.org/citation.cfm?id=368443>} interpolation and U{caching 

874 <https://GeographicLib.SourceForge.io/C++/doc/geoid.html#geoidcache>} 

875 in pure Python, transcoded from I{Karney}'s U{C++ class Geoid 

876 <https://GeographicLib.SourceForge.io/C++/doc/geoid.html#geoidinterp>}. 

877 

878 Use any of the geoid U{egm84-, egm96- or egm2008-*.pgm 

879 <https://GeographicLib.SourceForge.io/C++/doc/geoid.html#geoidinst>} 

880 datasets. 

881 ''' 

882 _C0 = _F(372), _F(240), _F(372) # n, _ and s common denominators 

883 # matrices c3n_, c3, c3s_, transposed from GeographicLib/Geoid.cpp 

884 _C3 = ((_T(0, 0, 62, 124, 124, 62, 0, 0, 0, 0, 0, 0), 

885 _T0s12, 

886 _T(-131, 7, -31, -62, -62, -31, 45, 216, 156, -45, -55, -7), 

887 _T0s12, 

888 _T(138, -138, 0, 0, 0, 0, -183, 33, 153, -3, 48, -48), # PYCHOK indent 

889 _T(144, 42, -62, -124, -124, -62, -9, 87, 99, 9, 42, -42), 

890 _T0s12, 

891 _T(0, 0, 0, 0, 0, 0, 93, -93, -93, 93, 0, 0), 

892 _T(-102, 102, 0, 0, 0, 0, 18, 12, -12, -18, -84, 84), 

893 _T(-31, -31, 31, 62, 62, 31, 0, -93, -93, 0, 31, 31)), # PYCHOK indent 

894 

895 (_T(9, -9, 9, 186, 54, -9, -9, 54, -54, 9, -9, 9), 

896 _T(-18, 18, -88, -42, 162, -32, 8, -78, 78, -8, 18, -18), 

897 _T(-88, 8, -18, -42, -78, 18, 18, 162, 78, -18, -32, -8), 

898 _T(0, 0, 90, -150, 30, 30, 30, -90, 90, -30, 0, 0), 

899 _T(96, -96, 96, -96, -24, 24, -96, -24, 144, -24, 24, -24), # PYCHOK indent 

900 _T(90, 30, 0, -150, -90, 0, 0, 30, 90, 0, 30, -30), 

901 _T(0, 0, -20, 60, -60, 20, -20, 60, -60, 20, 0, 0), 

902 _T(0, 0, -60, 60, 60, -60, 60, -60, -60, 60, 0, 0), 

903 _T(-60, 60, 0, 60, -60, 0, 0, 60, -60, 0, -60, 60), 

904 _T(-20, -20, 0, 60, 60, 0, 0, -60, -60, 0, 20, 20)), 

905 

906 (_T(18, -18, 36, 210, 162, -36, 0, 0, 0, 0, -18, 18), # PYCHOK indent 

907 _T(-36, 36, -165, 45, 141, -21, 0, 0, 0, 0, 36, -36), 

908 _T(-122, -2, -27, -111, -75, 27, 62, 124, 124, 62, -64, 2), 

909 _T(0, 0, 93, -93, -93, 93, 0, 0, 0, 0, 0, 0), 

910 _T(120, -120, 147, -57, -129, 39, 0, 0, 0, 0, 66, -66), # PYCHOK indent 

911 _T(135, 51, -9, -192, -180, 9, 31, 62, 62, 31, 51, -51), 

912 _T0s12, 

913 _T(0, 0, -93, 93, 93, -93, 0, 0, 0, 0, 0, 0), 

914 _T(-84, 84, 18, 12, -12, -18, 0, 0, 0, 0, -102, 102), 

915 _T(-31, -31, 0, 93, 93, 0, -31, -62, -62, -31, 31, 31))) 

916 

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

918 _T(1, 0), 

919 _T(0, 1), 

920 _T(1, 1)) 

921 

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

923 _T( 1, -1), 

924 _T(-1, 0), 

925 _T( 0, 0), 

926 _T( 1, 0), 

927 _T( 2, 0), 

928 _T(-1, 1), 

929 _T( 0, 1), 

930 _T( 1, 1), 

931 _T( 2, 1), 

932 _T( 0, 2), 

933 _T( 1, 2)) 

934 

935 _endian = '>H' # struct.unpack 1 ushort (big endian, unsigned short) 

936 _4endian = '>4H' # struct.unpack 4 ushorts 

937 _Rendian = NN # struct.unpack a row of ushorts 

938# _highest = (-8.4, 147.367, 85.839) if egm2008-1.pgm else ( 

939# (-8.167, 147.25, 85.422) if egm96-5.pgm else 

940# (-4.5, 148.75, 81.33)) # egm84-15.pgm 

941# _lowest = (4.7, 78.767, -106.911) if egm2008-1.pgm else ( 

942# (4.667, 78.833, -107.043) if egm96-5.pgm else 

943# (4.75, 79.25, -107.34)) # egm84-15.pgm 

944 _mean = _F(-1.317) # from egm2008-1, -1.438 egm96-5, -0.855 egm84-15 

945 _nBytes = None # not applicable 

946 _nterms = len(_C3[0]) # columns length, number of row 

947 _smooth = None # not applicable 

948 _stdev = _F(29.244) # from egm2008-1, 29.227 egm96-5, 29.183 egm84-15 

949 _u2B = _calcsize(_endian) # pixelsize_ in bytes 

950 _4u2B = _calcsize(_4endian) # 4 pixelsize_s in bytes 

951 _Ru2B = 0 # row of pixelsize_s in bytes 

952 _yxH = () # cache (y, x) indices 

953 _yxHt = () # cached 4- or 10-tuple for _ev2H resp. _ev3H 

954 _yx_hits = 0 # cache hits 

955 

956 def __init__(self, egm_pgm, crop=None, datum=None, # WGS84 

957 kind=3, **name_smooth): 

958 '''New L{GeoidKarney} interpolator. 

959 

960 @arg egm_pgm: An U{EGM geoid dataset<https://GeographicLib.SourceForge.io/ 

961 C++/doc/geoid.html#geoidinst>} file name (C{egm*.pgm}), see 

962 note below. 

963 @kwarg crop: Optional box to limit geoid locations, a 4-tuple (C{south, 

964 west, north, east}), 2-tuple (C{(south, west), (north, 

965 east)}) or 2, in C{degrees90} lat- and C{degrees180} 

966 longitudes or a 2-tuple (C{LatLonSW, LatLonNE}) of 

967 C{LatLon} instances. 

968 @kwarg datum: Optional grid datum (C{Datum}, L{Ellipsoid}, L{Ellipsoid2} 

969 or L{a_f2Tuple}), default C{WGS84}. 

970 @kwarg kind: Interpolation order (C{int}), 2 for C{bilinear} or 3 

971 for C{cubic}. 

972 @kwarg name_smooth: Optional geoid C{B{name}=NN} (C{str}) and UNSUPPORTED 

973 keyword argument C{B{smooth}}, use C{B{smooth}=None} to ignore. 

974 

975 @raise GeoidError: EGM dataset B{C{egm_pgm}} issue or invalid 

976 B{C{crop}}, B{C{kind}} or B{C{smooth}}. 

977 

978 @raise TypeError: Invalid B{C{datum}}. 

979 

980 @see: Class L{GeoidPGM} and function L{egmGeoidHeights}. 

981 

982 @note: Geoid file B{C{egm_pgm}} remains open and must be closed 

983 by calling the C{close} method or by using this instance 

984 in a C{with B{GeoidKarney}(...) as ...} context. 

985 ''' 

986 smooth, name = _xkwds_pop2(name_smooth, smooth=None) 

987 if smooth is not None: 

988 raise GeoidError(smooth=smooth, txt_not_=_supported_) 

989 

990 if kind in (2,): 

991 self._evH = self._ev2H 

992 elif kind not in (3,): 

993 raise GeoidError(kind=kind) 

994 

995 self._egm = g = self._open(egm_pgm, datum, kind, _name__(**name), None) 

996 self._pgm = p = _PGM(g, pgm=egm_pgm, itemsize=self.u2B, sizeB=self.sizeB) 

997 

998 self._Rendian = self._4endian.replace(_4_, str(p.nlon)) 

999 self._Ru2B = _calcsize(self._Rendian) 

1000 

1001 self._knots = p.knots # grid knots 

1002 self._lon_of = float(p.flon) # forward offset 

1003 self._lon_og = float(p.glon) # reverse offset 

1004 # set earth (lat, lon) limits (s, w, n, e) 

1005 self._lat_lo, self._lon_lo, \ 

1006 self._lat_hi, self._lon_hi = self._swne(crop if crop else p.crop4) 

1007 self._cropped = True if crop else False 

1008 

1009 def __call__(self, *llis, **wrap_H): 

1010 '''Interpolate the geoid height for one or several locations. 

1011 

1012 @arg llis: One or more locations (C{LatLon}s), all positional. 

1013 @kwarg wrap_H: Keyword arguments C{B{wrap}=False, B{H}=False}. 

1014 If C{B{wrap} is True}, wrap or I{normalize} all 

1015 B{C{llis}} locations (C{bool}). If C{B{H} is True}, 

1016 return the I{orthometric} height instead of the 

1017 I{geoid} height at each location (C{bool}). 

1018 

1019 @return: A single interpolated geoid (or orthometric) height 

1020 (C{float}) or a list or tuple of interpolated geoid 

1021 (or orthometric) heights (C{float}s). 

1022 

1023 @raise GeoidError: Insufficient number of B{C{llis}}, an 

1024 invalid B{C{lli}} or the C{egm*.pgm} 

1025 geoid file is closed. 

1026 

1027 @raise RangeError: An B{C{lli}} is outside this geoid's lat- 

1028 or longitude range. 

1029 

1030 @note: To obtain I{orthometric} heights, each B{C{llis}} 

1031 location must have an ellipsoid C{height} or C{h} 

1032 attribute, otherwise C{height=0} is used. 

1033 

1034 @see: Function L{pygeodesy.heightOrthometric}. 

1035 ''' 

1036 return self._called(llis, False, **wrap_H) 

1037 

1038 def _c0c3v(self, y, x): 

1039 # get the common denominator, the 10x12 cubic matrix and 

1040 # the 12 cubic v-coefficients around geoid index (y, x) 

1041 p = self._pgm 

1042 if 0 < x < (p.nlon - 2) and 0 < y < (p.nlat - 2): 

1043 # read 4x4 ushorts, drop the 4 corners 

1044 g = self._egm 

1045 e = self._4endian 

1046 n = self._4u2B 

1047 R = self._Ru2B 

1048 

1049 b = self._seek(y - 1, x - 1) 

1050 v = _unpack(e, g.read(n))[1:3] 

1051 b += R 

1052 g.seek(b, _SEEK_SET) 

1053 v += _unpack(e, g.read(n)) 

1054 b += R 

1055 g.seek(b, _SEEK_SET) 

1056 v += _unpack(e, g.read(n)) 

1057 b += R 

1058 g.seek(b, _SEEK_SET) 

1059 v += _unpack(e, g.read(n))[1:3] 

1060 j = 1 

1061 

1062 else: # likely some wrapped y and/or x's 

1063 v = self._raws(y, x, GeoidKarney._CM) 

1064 j = 0 if y < 1 else (1 if y < (p.nlat - 2) else 2) 

1065 

1066 return GeoidKarney._C0[j], GeoidKarney._C3[j], v 

1067 

1068 @Property_RO 

1069 def dtype(self): 

1070 '''Get the geoid's grid data type (C{str}). 

1071 ''' 

1072 return 'ushort' 

1073 

1074 def _ev(self, lat, lon): # PYCHOK expected 

1075 # interpolate the geoid height at grid (lat, lon) 

1076 fy, fx = self._g2yx2(lat, lon) 

1077 y, x = int(floor(fy)), int(floor(fx)) 

1078 fy -= y 

1079 fx -= x 

1080 H = self._evH(fy, fx, y, x) # ._ev3H or ._ev2H 

1081 H *= self._pgm.Scale # H.fmul(self._pgm.Scale) 

1082 H += self._pgm.Offset # H.fadd(self._pgm.Offset) 

1083 return H.fsum() 

1084 

1085 def _ev2H(self, fy, fx, *yx): 

1086 # compute the bilinear 4-tuple and interpolate raw H 

1087 if self._yxH == yx: 

1088 t = self._yxHt 

1089 self._yx_hits += 1 

1090 else: 

1091 y, x = self._yxH = yx 

1092 self._yxHt = t = self._raws(y, x, GeoidKarney._BT) 

1093 v = _1_0, -fx, fx 

1094 H = Fdot(v, t[0], t[0], t[1]).fmul(_1_0 - fy) # c = a * (1 - fy) 

1095 H += Fdot(v, t[2], t[2], t[3]).fmul(fy) # c += b * fy 

1096 return H 

1097 

1098 def _ev3H(self, fy, fx, *yx): 

1099 # compute the cubic 10-tuple and interpolate raw H 

1100 if self._yxH == yx: 

1101 t = self._yxHt 

1102 self._yx_hits += 1 

1103 else: 

1104 self._yxH = yx 

1105 c0, c3, v = self._c0c3v(*yx) 

1106 t = [fdot(v, *c3[i]) / c0 for i in range(self._nterms)] 

1107 self._yxHt = t = tuple(t) 

1108 # GeographicLib/Geoid.cpp Geoid::height(lat, lon) ... 

1109 # real h = t[0] + fx * (t[1] + fx * (t[3] + fx * t[6])) + 

1110 # fy * (t[2] + fx * (t[4] + fx * t[7]) + 

1111 # fy * (t[5] + fx * t[8] + fy * t[9])); 

1112 v = _1_0, fx, fy 

1113 H = Fdot(v, t[5], t[8], t[9]) 

1114 H *= fy 

1115 H += Fhorner(fx, t[2], t[4], t[7]) 

1116 H *= fy 

1117 H += Fhorner(fx, t[0], t[1], t[3], t[6]) 

1118 return H 

1119 

1120 _evH = _ev3H # overriden for kind == 2 

1121 

1122 def _g2ll2(self, lat, lon): 

1123 # convert grid (lat, lon) to earth (lat, lon), uncropped 

1124 while lon > _180_0: 

1125 lon -= _360_0 

1126 return lat, lon 

1127 

1128 def _g2yx2(self, lat, lon): 

1129 # convert grid (lat, lon) to grid (y, x) indices 

1130 p = self._pgm 

1131 # note, slat = +90, rlat < 0 makes y >=0 

1132 return ((lat - p.slat) * p.rlat), ((lon - p.wlon) * p.rlon) 

1133 

1134 def _gyx2g2(self, y, x): 

1135 # convert grid (y, x) indices to grid (lat, lon) 

1136 p = self._pgm 

1137 return (p.slat + p.dlat * y), (p.wlon + p.dlon * x) 

1138 

1139 def height(self, lats, lons, **wrap): 

1140 '''Interpolate the geoid height for one or several lat-/longitudes. 

1141 

1142 @arg lats: Latitude or latitudes (C{degrees} or C{degrees}s). 

1143 @arg lons: Longitude or longitudes (C{degrees} or C{degrees}s). 

1144 @kwarg wrap: If C{True}, wrap or I{normalize} all B{C{lats}} 

1145 and B{C{lons}} locations (C{bool}). 

1146 

1147 @return: A single interpolated geoid height (C{float}) or a 

1148 list of interpolated geoid heights (C{float}s). 

1149 

1150 @raise GeoidError: Insufficient or non-matching number of 

1151 B{C{lats}} and B{C{lons}} or the C{egm*.pgm} 

1152 geoid file is closed. 

1153 

1154 @raise RangeError: A B{C{lat}} or B{C{lon}} is outside this 

1155 geoid's lat- or longitude range. 

1156 ''' 

1157 return _height_called(self, lats, lons, Error=GeoidError, **wrap) 

1158 

1159 @Property_RO 

1160 def _highest_ltd(self): 

1161 '''(INTERNAL) Cache for L{highest} mesthod. 

1162 ''' 

1163 return self._llh3minmax(True, -12, -4) 

1164 

1165 def highest(self, LatLon=None, full=False): # PYCHOK full 

1166 '''Return the location and largest height of this geoid. 

1167 

1168 @kwarg LatLon: Optional class to return the location and height 

1169 (C{LatLon}) or C{None}. 

1170 @kwarg full: Search the full or limited latitude range (C{bool}). 

1171 

1172 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

1173 lon, height)} otherwise a B{C{LatLon}} instance 

1174 with the lat-, longitude and geoid height at the 

1175 highest grid location. 

1176 ''' 

1177 llh = self._highest if full or self.cropped else self._highest_ltd 

1178 return self._llh3LL(llh, LatLon) 

1179 

1180 def _lat2y2(self, lat2): 

1181 # convert earth lat(s) to min and max grid y indices 

1182 ys, m = [], self._pgm.nlat - 1 

1183 for lat in lat2: 

1184 y, _ = self._g2yx2(*self._ll2g2(lat, 0)) 

1185 ys.append(max(min(int(y), m), 0)) 

1186 return min(ys), max(ys) + 1 

1187 

1188 def _ll2g2(self, lat, lon): 

1189 # convert earth (lat, lon) to grid (lat, lon), uncropped 

1190 while lon < 0: 

1191 lon += _360_0 

1192 return lat, lon 

1193 

1194 def _llh3minmax(self, highest=True, *lat2): 

1195 # find highest or lowest, takes 10+ secs for egm2008-1.pgm geoid 

1196 # (Python 2.7.16, macOS 10.13.6 High Sierra, iMac 3 GHz Core i3) 

1197 y = x = 0 

1198 h = self._raw(y, x) 

1199 if highest: 

1200 for j, r in self._raw2(*lat2): 

1201 m = max(r) 

1202 if m > h: 

1203 h, y, x = m, j, r.index(m) 

1204 else: # lowest 

1205 for j, r in self._raw2(*lat2): 

1206 m = min(r) 

1207 if m < h: 

1208 h, y, x = m, j, r.index(m) 

1209 h *= self._pgm.Scale 

1210 h += self._pgm.Offset 

1211 return self._g2ll2(*self._gyx2g2(y, x)) + (h,) 

1212 

1213 @Property_RO 

1214 def _lowest_ltd(self): 

1215 '''(INTERNAL) Cache for L{lowest}. 

1216 ''' 

1217 return self._llh3minmax(False, 0, 8) 

1218 

1219 def lowest(self, LatLon=None, full=False): # PYCHOK full 

1220 '''Return the location and lowest height of this geoid. 

1221 

1222 @kwarg LatLon: Optional class to return the location and height 

1223 (C{LatLon}) or C{None}. 

1224 @kwarg full: Search the full or limited latitude range (C{bool}). 

1225 

1226 @return: If B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, 

1227 lon, height)} otherwise a B{C{LatLon}} instance 

1228 with the lat-, longitude and geoid height of the 

1229 lowest grid location. 

1230 ''' 

1231 llh = self._lowest if full or self.cropped else self._lowest_ltd 

1232 return self._llh3LL(llh, LatLon) 

1233 

1234 def _raw(self, y, x): 

1235 # get the ushort geoid height at geoid index (y, x), 

1236 # like GeographicLib/Geoid.hpp real rawval(is, iy) 

1237 p = self._pgm 

1238 if x < 0: 

1239 x += p.nlon 

1240 elif x >= p.nlon: 

1241 x -= p.nlon 

1242 h = p.nlon // 2 

1243 if y < 0: 

1244 y = -y 

1245 elif y >= p.nlat: 

1246 y = (p.nlat - 1) * 2 - y 

1247 else: 

1248 h = 0 

1249 x += h if x < h else -h 

1250 self._seek(y, x) 

1251 h = _unpack(self._endian, self._egm.read(self._u2B)) 

1252 return h[0] 

1253 

1254 def _raws(self, y, x, ijs): 

1255 # get bilinear 4-tuple or 10x12 cubic matrix 

1256 return tuple(self._raw(y + j, x + i) for i, j in ijs) 

1257 

1258 def _raw2(self, *lat2): 

1259 # yield a 2-tuple (y, ushorts) for each row or for 

1260 # the rows between two (or more) earth lat values 

1261 p = self._pgm 

1262 g = self._egm 

1263 e = self._Rendian 

1264 n = self._Ru2B 

1265 # min(lat2) <= lat <= max(lat2) or 0 <= y < p.nlat 

1266 s, t = self._lat2y2(lat2) if lat2 else (0, p.nlat) 

1267 self._seek(s, 0) # to start of row s 

1268 for y in range(s, t): 

1269 yield y, _unpack(e, g.read(n)) 

1270 

1271 def _seek(self, y, x): 

1272 # position geoid to grid index (y, x) 

1273 p, g = self._pgm, self._egm 

1274 if g: 

1275 b = p.skip + (y * p.nlon + x) * self._u2B 

1276 g.seek(b, _SEEK_SET) 

1277 return b # position 

1278 raise GeoidError('closed file', txt=repr(p.egm)) # IOError 

1279 

1280 

1281class GeoidPGM(_GeoidBase): 

1282 '''Geoid height interpolator for I{Karney}'s U{GeographicLib Earth 

1283 Gravitational Model (EGM)<https://GeographicLib.SourceForge.io/C++/doc/ 

1284 geoid.html>} geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/ 

1285 C++/doc/geoid.html#geoidinst>} datasets but based on C{SciPy} 

1286 U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/reference/ 

1287 generated/scipy.interpolate.RectBivariateSpline.html>} or 

1288 U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/ 

1289 scipy.interpolate.interp2d.html>} interpolation. 

1290 

1291 Use any of the U{egm84-, egm96- or egm2008-*.pgm 

1292 <https://GeographicLib.SourceForge.io/C++/doc/geoid.html#geoidinst>} 

1293 datasets. However, unless cropped, an entire C{egm*.pgm} dataset 

1294 is loaded into the C{SciPy} U{RectBivariateSpline<https://docs.SciPy.org/ 

1295 doc/scipy/reference/generated/scipy.interpolate.RectBivariateSpline.html>} 

1296 or U{interp2d<https://docs.SciPy.org/doc/scipy/reference/generated/ 

1297 scipy.interpolate.interp2d.html>} interpolator and converted from 

1298 2-byte C{int} to 8-byte C{dtype float64}. Therefore, internal memory 

1299 usage is 4x the U{egm*.pgm<https://GeographicLib.SourceForge.io/C++/doc/ 

1300 geoid.html#geoidinst>} file size and may exceed the available memory, 

1301 especially with 32-bit Python, see properties C{.nBytes} and C{.sizeB}. 

1302 ''' 

1303 _cropped = False 

1304 _endian = '>u2' 

1305 

1306 def __init__(self, egm_pgm, crop=None, datum=None, # WGS84 

1307 kind=3, smooth=0, **name): 

1308 '''New L{GeoidPGM} interpolator. 

1309 

1310 @arg egm_pgm: An U{EGM geoid dataset<https://GeographicLib.SourceForge.io/ 

1311 C++/doc/geoid.html#geoidinst>} file name (C{egm*.pgm}). 

1312 @kwarg crop: Optional box to crop B{C{egm_pgm}}, a 4-tuple (C{south, west, 

1313 north, east}) or 2-tuple (C{(south, west), (north, east)}), 

1314 in C{degrees90} lat- and C{degrees180} longitudes or a 

1315 2-tuple (C{LatLonSW, LatLonNE}) of C{LatLon} instances. 

1316 @kwarg datum: Optional grid datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} 

1317 or L{a_f2Tuple}), default C{WGS84}. 

1318 @kwarg kind: C{scipy.interpolate} order (C{int}), use 1..5 for 

1319 U{RectBivariateSpline<https://docs.SciPy.org/doc/scipy/ 

1320 reference/generated/scipy.interpolate.RectBivariateSpline.html>}, 

1321 -2 for U{interp2d linear<https://docs.SciPy.org/doc/scipy/ 

1322 reference/generated/scipy.interpolate.interp2d.html>}, -3 

1323 for C{interp2d cubic} or -5 for C{interp2d quintic}. 

1324 @kwarg smooth: Smoothing factor for U{RectBivariateSpline 

1325 <https://docs.SciPy.org/doc/scipy/reference/generated/ 

1326 scipy.interpolate.RectBivariateSpline.html>} 

1327 only (C{int}). 

1328 @kwarg name: Optional geoid C{B{name}=NN} (C{str}). 

1329 

1330 @raise GeoidError: EGM dataset B{C{egm_pgm}} issue or invalid B{C{crop}}, 

1331 B{C{kind}} or B{C{smooth}}. 

1332 

1333 @raise ImportError: Package C{numpy} or C{scipy} not found or not installed. 

1334 

1335 @raise LenError: EGM dataset B{C{egm_pgm}} axis mismatch. 

1336 

1337 @raise SciPyError: A C{RectBivariateSpline} or C{inter2d} issue. 

1338 

1339 @raise SciPyWarning: A C{RectBivariateSpline} or C{inter2d} 

1340 warning as exception. 

1341 

1342 @raise TypeError: Invalid B{C{datum}} or unexpected argument. 

1343 

1344 @note: C{scipy.interpolate.interp2d} has been C{DEPRECATED}, specify 

1345 C{B{kind}=1..5} for C{scipy.interpolate.RectBivariateSpline}. 

1346 

1347 @note: The U{GeographicLib egm*.pgm<https://GeographicLib.SourceForge.io/ 

1348 C++/doc/geoid.html#geoidinst>} file sizes are based on a 2-byte 

1349 C{int} height converted to 8-byte C{dtype float64} for C{scipy} 

1350 interpolators. Therefore, internal memory usage is 4 times the 

1351 C{egm*.pgm} file size and may exceed the available memory, 

1352 especially with 32-bit Python. To reduce memory usage, set 

1353 keyword argument B{C{crop}} to the region of interest. For example 

1354 C{B{crop}=(20, -125, 50, -65)} covers the U{conterminous US<https:// 

1355 www.NGS.NOAA.gov/GEOID/GEOID12B/maps/GEOID12B_CONUS_grids.png>} 

1356 (CONUS), less than 3% of the entire C{egm2008-1.pgm} dataset. 

1357 

1358 @see: Class L{GeoidKarney} and function L{egmGeoidHeights}. 

1359 ''' 

1360 np = self.numpy 

1361 self._u2B = np.dtype(self.endian).itemsize 

1362 

1363 g = self._open(egm_pgm, datum, kind, _name__(**name), smooth) 

1364 self._pgm = p = _PGM(g, pgm=egm_pgm, itemsize=self.u2B, sizeB=self.sizeB) 

1365 if crop: 

1366 g = p._cropped(g, abs(kind) + 1, *self._swne(crop)) 

1367 if _MODS.internals._version2(np.__version__) < (1, 9): 

1368 g = open(g.name, _rb_) # reopen tempfile for numpy 1.8.0- 

1369 self._cropped = True 

1370 try: 

1371 # U{numpy dtype formats are different from Python struct formats 

1372 # <https://docs.SciPy.org/doc/numpy-1.15.0/reference/arrays.dtypes.html>} 

1373 # read all heights, skipping the PGM header lines, converted to float 

1374 hs = self._load(g, self.endian, p.knots, p.skip).reshape(p.nlat, p.nlon) * p.Scale 

1375 if p.Offset: # offset 

1376 hs = p.Offset + hs 

1377 if p.dlat < 0: # flip the rows 

1378 hs = np.flipud(hs) 

1379 _GeoidBase.__init__(self, hs, p) 

1380 except Exception as x: 

1381 raise _SciPyIssue(x, _in_, repr(egm_pgm)) 

1382 finally: 

1383 g.close() 

1384 

1385 def _g2ll2(self, lat, lon): 

1386 # convert grid (lat, lon) to earth (lat, lon), un-/cropped 

1387 if self._cropped: 

1388 lon -= self._lon_of 

1389 else: 

1390 while lon > _180_0: 

1391 lon -= _360_0 

1392 return lat, lon 

1393 

1394 def _ll2g2(self, lat, lon): 

1395 # convert earth (lat, lon) to grid (lat, lon), un-/cropped 

1396 if self._cropped: 

1397 lon += self._lon_of 

1398 else: 

1399 while lon < 0: 

1400 lon += _360_0 

1401 return lat, lon 

1402 

1403 if _FOR_DOCS: 

1404 __call__ = _GeoidBase.__call__ 

1405 height = _GeoidBase.height 

1406 

1407 

1408class _Gpars(_Named): 

1409 '''(INTERNAL) Basic geoid parameters. 

1410 ''' 

1411 # interpolator parameters 

1412 dlat = 0 # +/- latitude resolution in C{degrees} 

1413 dlon = 0 # longitude resolution in C{degrees} 

1414 nlat = 1 # number of latitude knots (C{int}) 

1415 nlon = 0 # number of longitude knots (C{int}) 

1416 rlat = 0 # +/- latitude resolution in C{float}, 1 / .dlat 

1417 rlon = 0 # longitude resolution in C{float}, 1 / .dlon 

1418 slat = 0 # nothern- or southern most latitude (C{degrees90}) 

1419 wlon = 0 # western-most longitude in Eastern lon (C{degrees360}) 

1420 

1421 flon = 0 # forward, earth to grid longitude offset 

1422 glon = 0 # reverse, grid to earth longitude offset 

1423 

1424 knots = 0 # number of knots, nlat * nlon (C{int}) 

1425 skip = 0 # header bytes to skip (C{int}) 

1426 

1427 def __repr__(self): 

1428 t = _COMMASPACE_.join(pairs((a, getattr(self, a)) for 

1429 a in dir(self.__class__) 

1430 if a[:1].isupper())) 

1431 return _COLONSPACE_(self, t) 

1432 

1433 def __str__(self): 

1434 return Fmt.PAREN(self.classname, repr(self.name)) 

1435 

1436 

1437class _PGM(_Gpars): 

1438 '''(INTERNAL) Parse an C{egm*.pgm} geoid dataset file. 

1439 

1440 # Geoid file in PGM format for the GeographicLib::Geoid class 

1441 # Description WGS84 EGM96, 5-minute grid 

1442 # URL https://Earth-Info.NGA.mil/GandG/wgs84/gravitymod/egm96/egm96.html 

1443 # DateTime 2009-08-29 18:45:03 

1444 # MaxBilinearError 0.140 

1445 # RMSBilinearError 0.005 

1446 # MaxCubicError 0.003 

1447 # RMSCubicError 0.001 

1448 # Offset -108 

1449 # Scale 0.003 

1450 # Origin 90N 0E 

1451 # AREA_OR_POINT Point 

1452 # Vertical_Datum WGS84 

1453 <width> <height> 

1454 <pixel> 

1455 ... 

1456 ''' 

1457 crop4 = () # 4-tuple (C{south, west, north, east}). 

1458 egm = None 

1459 glon = 180 # reverse offset, uncropped 

1460# pgm = NN # name 

1461 sizeB = 0 

1462 u2B = 2 # item size of grid height (C{int}). 

1463 

1464 @staticmethod 

1465 def _llstr2floats(latlon): 

1466 # llstr to (lat, lon) floats 

1467 lat, lon = latlon.split() 

1468 return _MODS.dms.parseDMS2(lat, lon) 

1469 

1470 # PGM file attributes, CamelCase but not .istitle() 

1471 AREA_OR_POINT = str 

1472 DateTime = str 

1473 Description = str # 'WGS84 EGM96, 5-minute grid' 

1474 Geoid = str # 'file in PGM format for the GeographicLib::Geoid class' 

1475 MaxBilinearError = float 

1476 MaxCubicError = float 

1477 Offset = float 

1478 Origin = _llstr2floats 

1479 Pixel = 0 

1480 RMSBilinearError = float 

1481 RMSCubicError = float 

1482 Scale = float 

1483 URL = str # 'https://Earth-Info.NGA.mil/GandG/wgs84/...' 

1484 Vertical_Datum = str 

1485 

1486 def __init__(self, g, pgm=NN, itemsize=0, sizeB=0): # MCCABE 22 

1487 '''(INTERNAL) New C{_PGM} parsed C{egm*.pgm} geoid dataset. 

1488 ''' 

1489 self.name = pgm # geoid file name 

1490 if itemsize: 

1491 self._u2B = itemsize 

1492 if sizeB: 

1493 self.sizeB = sizeB 

1494 

1495 t = g.readline() # make sure newline == '\n' 

1496 if t != b'P5\n' and t.strip() != b'P5': 

1497 raise self._Errorf(_format_, _header_, t) 

1498 

1499 while True: # read all # Attr ... lines, 

1500 try: # ignore empty ones or comments 

1501 t = g.readline().strip() 

1502 if t.startswith(_bHASH_): 

1503 t = t.lstrip(_bHASH_).lstrip() 

1504 a, v = map(_ub2str, t.split(None, 1)) 

1505 f = getattr(_PGM, a, None) 

1506 if callable(f) and a[:1].isupper(): 

1507 setattr(self, a, f(v)) 

1508 elif t: 

1509 break 

1510 except (TypeError, ValueError): 

1511 raise self._Errorf(_format_, 'Attr', t) 

1512 else: # should never get here 

1513 raise self._Errorf(_format_, _header_, g.tell()) 

1514 

1515 try: # must be (even) width and (odd) height 

1516 nlon, nlat = map(int, t.split()) 

1517 if nlon < 2 or nlon > (360 * 60) or isodd(nlon) or \ 

1518 nlat < 2 or nlat > (181 * 60) or not isodd(nlat): 

1519 raise ValueError 

1520 except (TypeError, ValueError): 

1521 raise self._Errorf(_format_, _SPACE_(_width_, _height_), t) 

1522 

1523 try: # must be 16 bit pixel height 

1524 t = g.readline().strip() 

1525 self.Pixel = int(t) 

1526 if not 255 < self.Pixel < 65536: # >u2 or >H only 

1527 raise ValueError 

1528 except (TypeError, ValueError): 

1529 raise self._Errorf(_format_, 'pixel', t) 

1530 

1531 for a in dir(_PGM): # set undefined # Attr ... to None 

1532 if a[:1].isupper() and callable(getattr(self, a)): 

1533 setattr(self, a, None) 

1534 

1535 if self.Origin is None: 

1536 raise self._Errorf(_format_, 'Origin', self.Origin) 

1537 if self.Offset is None or self.Offset > 0: 

1538 raise self._Errorf(_format_, 'Offset', self.Offset) 

1539 if self.Scale is None or self.Scale < EPS: 

1540 raise self._Errorf(_format_, 'Scale', self.Scale) 

1541 

1542 self.skip = g.tell() 

1543 self.knots = nlat * nlon 

1544 

1545 self.nlat, self.nlon = nlat, nlon 

1546 self.slat, self.wlon = self.Origin 

1547 # note, negative .dlat and .rlat since rows 

1548 # are from .slat 90N down in decreasing lat 

1549 self.dlat, self.dlon = _180_0 / (1 - nlat), _360_0 / nlon 

1550 self.rlat, self.rlon = (1 - nlat) / _180_0, nlon / _360_0 

1551 

1552 # grid corners in earth (lat, lon), .slat = 90, .dlat < 0 

1553 n = float(self.slat) 

1554 s = n + self.dlat * (nlat - 1) 

1555 w = self.wlon - self.glon 

1556 e = w + self.dlon * nlon 

1557 self.crop4 = s, w, n, e 

1558 

1559 n = self.sizeB - self.skip 

1560 if n > 0 and n != (self.knots * self.u2B): 

1561 raise self._Errorf('%s(%s x %s != %s)', _assert_, nlat, nlon, n) 

1562 

1563 def _cropped(self, g, k1, south, west, north, east): # MCCABE 15 

1564 '''Crop the geoid to (south, west, north, east) box. 

1565 ''' 

1566 # flon offset for both west and east 

1567 f = 360 if west < 0 else 0 

1568 # earth (lat, lon) to grid indices (y, x), 

1569 # note y is decreasing, i.e. n < s 

1570 s, w = self._lle2yx2(south, west, f) 

1571 n, e = self._lle2yx2(north, east, f) 

1572 s += 1 # s > n 

1573 e += 1 # e > w 

1574 

1575 hi, wi = self.nlat, self.nlon 

1576 # handle special cases 

1577 if (s - n) > hi: 

1578 n, s = 0, hi # entire lat range 

1579 if (e - w) > wi: 

1580 w, e, f = 0, wi, 180 # entire lon range 

1581 if s == hi and w == n == 0 and e == wi: 

1582 return g # use entire geoid as-is 

1583 

1584 if (e - w) < k1 or (s - n) < (k1 + 1): 

1585 raise self._Errorf(_format_, 'swne', (north - south, east - west)) 

1586 

1587 if e > wi > w: # wrap around 

1588 # read w..wi and 0..e 

1589 r, p = (wi - w), (e - wi) 

1590 elif e > w: 

1591 r, p = (e - w), 0 

1592 else: 

1593 raise self._Errorf('%s(%s < %s)', _assert_, w, e) 

1594 

1595 # convert to bytes 

1596 r *= self.u2B 

1597 p *= self.u2B 

1598 q = wi * self.u2B # stride 

1599 # number of rows and cols to skip from 

1600 # the original (.slat, .wlon) origin 

1601 z = self.skip + (n * wi + w) * self.u2B 

1602 # sanity check 

1603 if r < 2 or p < 0 or q < 2 or z < self.skip \ 

1604 or z > self.sizeB: 

1605 raise self._Errorf(_format_, _assert_, (r, p, q, z)) 

1606 

1607 # can't use _BytesIO since numpy 

1608 # needs .fileno attr in .fromfile 

1609 t, c = 0, self._tmpfile() 

1610 # reading (s - n) rows, forward 

1611 for y in range(n, s): # PYCHOK y unused 

1612 g.seek(z, _SEEK_SET) 

1613 # Python 2 tmpfile.write returns None 

1614 t += c.write(g.read(r)) or r 

1615 if p: # wrap around to start of row 

1616 g.seek(-q, _SEEK_CUR) 

1617 # assert(g.tell() == (z - w * self.u2B)) 

1618 # Python 2 tmpfile.write returns None 

1619 t += c.write(g.read(p)) or p 

1620 z += q 

1621 c.flush() 

1622 g.close() 

1623 

1624 s -= n # nlat 

1625 e -= w # nlon 

1626 k = s * e # knots 

1627 z = k * self.u2B 

1628 if t != z: 

1629 raise self._Errorf('%s(%s != %s) %s', _assert_, t, z, self) 

1630 

1631 # update the _Gpars accordingly, note attributes 

1632 # .dlat, .dlon, .rlat and .rlon remain unchanged 

1633 self.slat += n * self.dlat 

1634 self.wlon += w * self.dlon 

1635 self.nlat = s 

1636 self.nlon = e 

1637 self.flon = self.glon = f 

1638 

1639 self.crop4 = south, west, north, east 

1640 self.knots = k 

1641 self.skip = 0 # no header lines in c 

1642 

1643 c.seek(0, _SEEK_SET) 

1644 # c = open(c.name, _rb_) # reopen for numpy 1.8.0- 

1645 return c 

1646 

1647 def _Errorf(self, fmt, *args): # PYCHOK no cover 

1648 t = fmt % args 

1649 e = self.pgm or NN 

1650 if e: 

1651 t = _SPACE_(t, _in_, repr(e)) 

1652 return PGMError(t) 

1653 

1654 def _lle2yx2(self, lat, lon, flon): 

1655 # earth (lat, lon) to grid indices (y, x) 

1656 # with .dlat decreasing from 90N .slat 

1657 lat -= self.slat 

1658 lon += flon - self.wlon 

1659 return (min(self.nlat - 1, max(0, int(lat * self.rlat))), 

1660 max(0, int(lon * self.rlon))) 

1661 

1662 def _tmpfile(self): 

1663 # create a tmpfile to hold the cropped geoid grid 

1664 try: 

1665 from tempfile import NamedTemporaryFile as tmpfile 

1666 except ImportError: # Python 2.7.16- 

1667 from os import tmpfile 

1668 t = _os_path.splitext(_os_path.basename(self.pgm))[0] 

1669 f = tmpfile(mode='w+b', prefix=t or 'egm') 

1670 f.seek(0, _SEEK_SET) # force overwrite 

1671 return f 

1672 

1673 @Property_RO 

1674 def pgm(self): 

1675 '''Get the geoid file name (C{str}). 

1676 ''' 

1677 return self.name 

1678 

1679 

1680class PGMError(GeoidError): 

1681 '''Issue parsing or cropping an C{egm*.pgm} geoid dataset. 

1682 ''' 

1683 pass 

1684 

1685 

1686def egmGeoidHeights(GeoidHeights_dat): 

1687 '''Generate geoid U{egm*.pgm<https://GeographicLib.SourceForge.io/ 

1688 C++/doc/geoid.html#geoidinst>} height tests from U{GeoidHeights.dat 

1689 <https://SourceForge.net/projects/geographiclib/files/testdata/>} 

1690 U{Test data for Geoids<https://GeographicLib.SourceForge.io/C++/doc/ 

1691 geoid.html#testgeoid>}. 

1692 

1693 @arg GeoidHeights_dat: The un-gz-ed C{GeoidHeights.dat} file 

1694 (C{str} or C{file} handle). 

1695 

1696 @return: For each test, yield a L{GeoidHeight5Tuple}C{(lat, lon, 

1697 egm84, egm96, egm2008)}. 

1698 

1699 @raise GeoidError: Invalid B{C{GeoidHeights_dat}}. 

1700 

1701 @note: Function L{egmGeoidHeights} is used to test the geoids 

1702 L{GeoidKarney} and L{GeoidPGM}, see PyGeodesy module 

1703 C{test/testGeoids.py}. 

1704 ''' 

1705 dat = GeoidHeights_dat 

1706 if isinstance(dat, bytes): 

1707 dat = _BytesIO(dat) 

1708 

1709 try: 

1710 dat.seek(0, _SEEK_SET) # reset 

1711 except AttributeError as x: 

1712 raise GeoidError(GeoidHeights_dat=type(dat), cause=x) 

1713 

1714 for t in dat.readlines(): 

1715 t = t.strip() 

1716 if t and not t.startswith(_bHASH_): 

1717 lat, lon, egm84, egm96, egm2008 = map(float, t.split()) 

1718 while lon > _180_0: # EasternLon to earth lon 

1719 lon -= _360_0 

1720 yield GeoidHeight5Tuple(lat, lon, egm84, egm96, egm2008) 

1721 

1722 

1723__all__ += _ALL_DOCS(_GeoidBase) 

1724 

1725if __name__ == '__main__': 

1726 

1727 from pygeodesy.internals import printf, _sys 

1728 

1729 _crop = () 

1730 _GeoidEGM = GeoidKarney 

1731 _kind = 3 

1732 

1733 geoids = _sys.argv[1:] 

1734 while geoids: 

1735 geoid = geoids.pop(0) 

1736 

1737 if '-crop'.startswith(geoid.lower()): 

1738 _crop = 20, -125, 50, -65 # CONUS 

1739 

1740 elif '-karney'.startswith(geoid.lower()): 

1741 _GeoidEGM = GeoidKarney 

1742 

1743 elif '-kind'.startswith(geoid.lower()): 

1744 _kind = int(geoids.pop(0)) 

1745 

1746 elif '-pgm'.startswith(geoid.lower()): 

1747 _GeoidEGM = GeoidPGM 

1748 

1749 elif geoid[-4:].lower() in ('.pgm',): 

1750 g = _GeoidEGM(geoid, crop=_crop, kind=_kind) 

1751 printf(g.toStr(), nt=1, nl=1) 

1752 printf(repr(g.pgm), nt=1) 

1753 # <https://GeographicLib.SourceForge.io/cgi-bin/GeoidEval>: 

1754 # The height of the EGM96 geoid at Timbuktu 

1755 # echo 16:46:33N 3:00:34W | GeoidEval 

1756 # => 28.7068 -0.02e-6 -1.73e-6 

1757 # The 1st number is the height of the geoid, the 2nd and 

1758 # 3rd are its slopes in northerly and easterly direction 

1759 t = 'Timbuktu %s' % (g,) 

1760 k = {'egm84-15.pgm': '31.2979', 

1761 'egm96-5.pgm': '28.7067', 

1762 'egm2008-1.pgm': '28.7880'}.get(g.name.lower(), '28.7880') 

1763 ll = _MODS.dms.parseDMS2('16:46:33N', '3:00:34W', sep=':') 

1764 for ll in (ll, (16.776, -3.009),): 

1765 try: 

1766 h, ll = g.height(*ll), fstr(ll, prec=6) 

1767 printf('%s.height(%s): %.4F vs %s', t, ll, h, k) 

1768 except (GeoidError, RangeError) as x: 

1769 printf(_COLONSPACE_(t, str(x))) 

1770 

1771 elif geoid[-4:].lower() in ('.bin',): 

1772 g = GeoidG2012B(geoid, kind=_kind) 

1773 printf(g.toStr()) 

1774 

1775 else: 

1776 raise GeoidError(grid=repr(geoid)) 

1777 

1778_I = int # PYCHOK unused _I 

1779del _intCs # trash ints cache 

1780 

1781# **) MIT License 

1782# 

1783# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1784# 

1785# Permission is hereby granted, free of charge, to any person obtaining a 

1786# copy of this software and associated documentation files (the "Software"), 

1787# to deal in the Software without restriction, including without limitation 

1788# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1789# and/or sell copies of the Software, and to permit persons to whom the 

1790# Software is furnished to do so, subject to the following conditions: 

1791# 

1792# The above copyright notice and this permission notice shall be included 

1793# in all copies or substantial portions of the Software. 

1794# 

1795# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1796# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1797# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1798# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1799# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1800# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1801# OTHER DEALINGS IN THE SOFTWARE. 

1802 

1803# <https://GeographicLib.SourceForge.io/cgi-bin/GeoidEval> 

1804# _lowerleft = -90, -179, -30.1500 # egm2008-1.pgm 

1805# _lowerleft = -90, -179, -29.5350 # egm96-5.pgm 

1806# _lowerleft = -90, -179, -29.7120 # egm84-15.pgm 

1807 

1808# _center = 0, 0, 17.2260 # egm2008-1.pgm 

1809# _center = 0, 0, 17.1630 # egm96-5.pgm 

1810# _center = 0, 0, 18.3296 # egm84-15.pgm 

1811 

1812# _upperright = 90, 180, 14.8980 # egm2008-1.pgm 

1813# _upperright = 90, 180, 13.6050 # egm96-5.pgm 

1814# _upperright = 90, 180, 13.0980 # egm84-15.pgm 

1815 

1816 

1817# % python3 -m pygeodesy.geoids [-Karney] ../testGeoids/egm*.pgm 

1818# 

1819# 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, 147.367, 85.839), lowest(4.7, 78.767, -106.911) 

1820# 

1821# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.001, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008', Vertical_Datum='WGS84' 

1822# 

1823# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.775833, -3.009444): 28.7881 vs 28.7880 

1824# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.776, -3.009): 28.7880 vs 28.7880 

1825# 

1826# 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(-4.5, 148.75, 81.33), lowest(4.75, 79.25, -107.34) 

1827# 

1828# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.018, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html', Vertical_Datum='WGS84' 

1829# 

1830# Timbuktu GeoidKarney('egm84-15.pgm').height(16.775833, -3.009444): 31.2983 vs 31.2979 

1831# Timbuktu GeoidKarney('egm84-15.pgm').height(16.776, -3.009): 31.2979 vs 31.2979 

1832# 

1833# 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(-8.167, 147.25, 85.422), lowest(4.667, 78.833, -107.043) 

1834# 

1835# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.005, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm96/egm96.html', Vertical_Datum='WGS84' 

1836# 

1837# Timbuktu GeoidKarney('egm96-5.pgm').height(16.775833, -3.009444): 28.7068 vs 28.7067 

1838# Timbuktu GeoidKarney('egm96-5.pgm').height(16.776, -3.009): 28.7067 vs 28.7067 

1839 

1840 

1841# % python3 -m pygeodesy.geoids -Karney ../testGeoids/egm*.pgm 

1842# 

1843# 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, 147.367, 85.839), lowest(4.7, 78.767, -106.911) 

1844# 

1845# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.001, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008', Vertical_Datum='WGS84' 

1846# 

1847# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.775833, -3.009444): 28.7881 vs 28.7880 

1848# Timbuktu GeoidKarney('egm2008-1.pgm').height(16.776, -3.009): 28.7880 vs 28.7880 

1849# 

1850# 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(-4.5, 148.75, 81.33), lowest(4.75, 79.25, -107.34) 

1851# 

1852# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.018, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html', Vertical_Datum='WGS84' 

1853# 

1854# Timbuktu GeoidKarney('egm84-15.pgm').height(16.775833, -3.009444): 31.2983 vs 31.2979 

1855# Timbuktu GeoidKarney('egm84-15.pgm').height(16.776, -3.009): 31.2979 vs 31.2979 

1856# 

1857# 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(-8.167, 147.25, 85.422), lowest(4.667, 78.833, -107.043) 

1858# 

1859# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.005, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm96/egm96.html', Vertical_Datum='WGS84' 

1860# 

1861# Timbuktu GeoidKarney('egm96-5.pgm').height(16.775833, -3.009444): 28.7068 vs 28.7067 

1862# Timbuktu GeoidKarney('egm96-5.pgm').height(16.776, -3.009): 28.7067 vs 28.7067 

1863 

1864 

1865# % python2 -m pygeodesy.geoids -PGM ../testGeoids/egm*.pgm 

1866# 

1867# 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) 

1868# 

1869# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.001, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008', Vertical_Datum='WGS84' 

1870# 

1871# Timbuktu GeoidPGM('egm2008-1.pgm').height(16.775833, -3.009444): 28.7881 vs 28.7880 

1872# Timbuktu GeoidPGM('egm2008-1.pgm').height(16.776, -3.009): 28.7880 vs 28.7880 

1873# 

1874# 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) 

1875# 

1876# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.018, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html', Vertical_Datum='WGS84' 

1877# 

1878# Timbuktu GeoidPGM('egm84-15.pgm').height(16.775833, -3.009444): 31.2979 vs 31.2979 

1879# Timbuktu GeoidPGM('egm84-15.pgm').height(16.776, -3.009): 31.2975 vs 31.2979 

1880# 

1881# 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) 

1882# 

1883# _PGM('../testGeoids/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=LatLon2Tuple(lat=90.0, lon=0.0), Pixel=65535, RMSBilinearError=0.005, RMSCubicError=0.001, Scale=0.003, URL='http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm96/egm96.html', Vertical_Datum='WGS84' 

1884# 

1885# Timbuktu GeoidPGM('egm96-5.pgm').height(16.775833, -3.009444): 28.7065 vs 28.7067 

1886# Timbuktu GeoidPGM('egm96-5.pgm').height(16.776, -3.009): 28.7064 vs 28.7067