content: # Description: General-purpose functions for personal use.
  # Author:      André Palóczy
  # E-mail:      paloczy@gmail.com
  
  __all__ = ['seasonal_avg',
             'seasonal_std',
             'deseason',
             'blkavg',
             'blkavgdir',
             'blkavgt',
             'blkapply',
             'stripmsk',
             'pydatetime2m_arr',
             'm2pydatetime_arr',
             'npdt2dt',
             'dt2sfloat',
             'doy2date',
             'flowfun',
             'cumsimp',
             'rot_vec',
             'avgdir',
             'lon180to360',
             'lon360to180',
             'bbox2ij',
             'xy2dist',
             'get_xtrackline',
             'get_arrdepth',
             'fpointsbox',
             'near',
             'near2',
             'mnear',
             'refine',
             'denan',
             'standardize',
             'linear_trend',
             'thomas',
             'point_in_poly',
             'get_mask_from_poly',
             'sphericalpolygon_area',
             'greatCircleBearing',
             'weim',
             'smoo2',
             'topo_slope',
             'curvature_geometric',
             'get_isobath',
             'angle_isobath',
             'isopyc_depth',
  		   'whiten_zero',
             'wind2stress',
             'gen_dates',
             'fmt_isobath',
             'float2latex',
             'mat2npz',
             'bb_map',
             'dots_dualcolor']
  
  from os import system
  import numpy as np
  import matplotlib.pyplot as plt
  import matplotlib
  from matplotlib import path
  from mpl_toolkits.basemap import Basemap
  from datetime import datetime, timedelta
  from dateutil import rrule, parser
  from scipy.io import loadmat, savemat
  from scipy import signal
  from scipy.signal import savgol_filter
  from glob import glob
  from netCDF4 import Dataset, num2date, date2num
  # from pandas import rolling_window # FIXME, new pandas way of doing this is, e.g., arr = Series(...).rolling(...).mean()
  from pandas import Timestamp
  from gsw import distance
  from pygeodesy import Datums, VincentyError
  from pygeodesy.ellipsoidalVincenty import LatLon as LatLon
  from pygeodesy.sphericalNvector import LatLon as LatLon_sphere
  
  
  def seasonal_avg(t, F):
      """
      USAGE
      -----
      F_seasonal = seasonal_avg(t, F)
  
      Calculates the seasonal average of variable F(t).
      Assumes 't' is a 'datetime.datetime' object.
      """
      tmo = np.array([ti.month for ti in t])
      ftmo = [tmo==mo for mo in range(1, 13)]
  
      return np.array([F[ft].mean() for ft in ftmo])
  
  
  def seasonal_std(t, F):
      """
      USAGE
      -----
      F_seasonal = seasonal_std(t, F)
  
      Calculates the seasonal standard deviation of variable F(t).
      Assumes 't' is a 'datetime.datetime' object.
      """
      tmo = np.array([ti.month for ti in t])
      ftmo = [tmo==mo for mo in range(1, 13)]
  
      return np.array([F[ft].std() for ft in ftmo])
  
  
  def deseason(t, F):
      """
      USAGE
      -----
      F_nonssn = deseason(t, F)
  
      Removes the seasonal signal of variable F(t).
      Assumes 't' is a 'datetime.datetime' object.
      Also assumes that F is sampled monthly and only for
      complete years (i.e., t.size is a multiple of 12).
      """
      Fssn = seasonal_avg(t, F)
      nyears = int(t.size/12)
      aux = np.array([])
      for n in range(nyears):
          aux = np.concatenate((aux, Fssn))
  
      return F - aux
  
  
  def blkavg(x, y, every=2):
      """
      Block-averages a variable y(x). Returns its block average
      and standard deviation and new x axis.
      """
      nx = x.size
      xblk, yblk, yblkstd = np.array([]), np.array([]), np.array([])
      for i in range(every, nx+every, every):
          yi = y[i-every:i]
          xblk = np.append(xblk, np.nanmean(x[i-every:i]))
          yblk = np.append(yblk, np.nanmean(yi))
          yblkstd = np.append(yblkstd, np.nanstd(yi))
  
      return xblk, yblk, yblkstd
  
  
  def blkavgdir(x, ydir, every=2, degrees=False, axis=None):
      """
      Block-averages a PERIODIC variable ydir(x). Returns its
      block average and new x axis.
      """
      nx = x.size
      xblk, yblk, yblkstd = np.array([]), np.array([]), np.array([])
      for i in range(every, nx+every, every):
          xblk = np.append(xblk, np.nanmean(x[i-every:i]))
          yblk = np.append(yblk, avgdir(ydir[i-every:i], degrees=degrees, axis=axis))
  
      return xblk, yblk
  
  
  def blkavgt(t, x, every=2):
      """
      Block-averages a variable x(t). Returns its block average
      and the new t axis.
      """
      nt = t.size
      units = 'days since 01-01-01'
      calendar = 'proleptic_gregorian'
      t = date2num(t, units=units, calendar=calendar)
      tblk, xblk = np.array([]), np.array([])
      for i in range(every, nt+every, every):
          xi = x[i-every:i]
          tblk = np.append(tblk, np.nanmean(t[i-every:i]))
          xblk = np.append(xblk, np.nanmean(xi))
  
      tblk = num2date(tblk, units=units, calendar=calendar)
      return tblk, xblk
  
  
  def blkapply(x, f, nblks, overlap=0, demean=False, detrend=False, verbose=True):
      """
      Divides array 'x' in 'nblks' blocks and applies function 'f' = f(x) on
      each block.
      """
      x = np.array(x)
      assert callable(f), "f must be a function"
  
      nx = x.size
      ni = int(nx/nblks)               # Number of data points in each chunk.
      y = np.zeros(ni)                 # Array that will receive each block.
      dn = int(round(ni - overlap*ni)) # How many indices to move forward with
                                       # each chunk (depends on the % overlap).
      # Demean/detrend the full record first (removes the lowest frequencies).
      # Then, also demean/detrend each block beffore applying f().
      if demean: x = x - x.mean()
      if detrend: x = signal.detrend(x, type='linear')
  
      n=0
      il, ir = 0, ni
      while ir<=nx:
          xn = x[il:ir]
          if demean: xn = xn - xn.mean()
          if detrend: xn = signal.detrend(xn, type='linear')
          y = y + f(xn) # Apply function and accumulate the current bock.
          il+=dn; ir+=dn
          n+=1
  
      y /= n         # Divide by number of blocks actually used.
      ncap = nx - il # Number of points left out at the end of array.
  
      if verbose:
          print("")
          print("Left last %d data points out (%.1f %% of all points)."%(ncap,100*ncap/nx))
          if overlap>0:
              print("")
              print("Intended %d blocks, but could fit %d blocks, with"%(nblks,n))
              print('overlap of %.1f %%, %d points per block.'%(100*overlap,dn))
          print("")
  
      return y
  
  
  def stripmsk(arr, mask_invalid=False):
      if mask_invalid:
          arr = np.ma.masked_invalid(arr)
      if np.ma.isMA(arr):
          msk = arr.mask
          arr = arr.data
          arr[msk] = np.nan
  
      return arr
  
  
  def pydatetime2m_arr(pydt_arr):
      pydt_arr = np.array(pydt_arr)
      secperyr = 86400.0
      timedt = timedelta(days=366)
      matdt = []
      for pydt in pydt_arr.tolist():
          m = pydt.toordinal() + timedt
          dfrac = pydt - datetime(pydt.year,pydt.month,pydt.day,0,0,0).seconds/secperyr
          matdt.append(m.toordinal() + dfrac)
  
      return np.array(matdt)
  
  
  def m2pydatetime_arr(mdatenum_arr):
      mdatenum_arr = np.array(mdatenum_arr)
      timedt = timedelta(days=366)
      pydt = []
      for mdt in mdatenum_arr.tolist():
          d = datetime.fromordinal(int(mdt))
          dfrac = timedelta(days=mdt%1) - timedt
          pydt.append(d + dfrac)
  
      return np.array(pydt)
  
  
  def npdt2dt(tnp):
      """
      USAGE
      -----
      t_datetime = npdt2dt(t_numpydatetime64)
  
      Convert an array of numpy.datetime64 timestamps to datetime.datetime.
      """
      return np.array([Timestamp(ti).to_pydatetime() for ti in tnp])
  
  
  def dt2sfloat(t):
      """
      USAGE
      -----
      t_float = dt2sfloat(t_datetime)
  
      Convert an array of datetime.datetime timestamps to an array of floats
      representing elapsed seconds since the first timestamp.
      """
      t = np.array(t)
      t0 = t[0]
  
      return np.array([(tn - t0).total_seconds() for tn in t])
  
  
  def doy2date(doy, year=2017):
      """
      USAGE
      -----
      t = doy2date(doy, year=2017)
  
      Convert an array `doy` of decimal yeardays to
      an array of datetime.datetime timestamps.
      """
      doy = np.array(doy)*86400 # [seconds/day].
      tunit = 'seconds since %d-01-01 00:00:00'%year
  
      return np.array([num2date(dn, tunit) for dn in doy])
  
  
  def flowfun(x, y, u, v, variable='psi', geographic=True):
  	"""
  	FLOWFUN  Computes the potential PHI and the streamfunction PSI
  	 of a 2-dimensional flow defined by the matrices of velocity
  	 components U and V, so that
  
  	       d(PHI)    d(PSI)          d(PHI)    d(PSI)
  	  u =  -----  -  ----- ,    v =  -----  +  -----
  	        dx        dy              dx        dy
  
  	P = FLOWFUN(x,y,u,v) returns an array P of the same size as u and v,
  	which can be the velocity potential (PHI) or the streamfunction (PSI)
  	Because these scalar fields are defined up to the integration constant,
  	their absolute values are such that PHI[0,0] = PSI[0,0] = 0.
  
  	For a potential (irrotational) flow  PSI = 0, and the Laplacian
  	of PSI is equal to the divergence of the velocity field.
  
  	A solenoidal (non-divergent) flow can be described by the
  	streamfunction alone, and the Laplacian of the streamfunction
  	is equal to the vorticity (curl) of the velocity field.
  
  	The units of the grid coordinates are assumed to be consistent
  	with the units of the velocity components, e.g., [m] and [m/s].
  
  	If variable=='psi', the streamfunction (PSI) is returned.
  
  	If variable=='phi', the velocity potential (PHI) is returned.
  
  	If geographic==True (default), (x,y) are assumed to be
  	(longitude,latitude) and are converted to meters before
  	computing (dx,dy).
  
  	If geographic==False, (x,y) are assumed to be in meters.
  
  	Uses function 'cumsimp()' (Simpson rule summation).
  
  	Author: Kirill K. Pankratov, March 7, 1994.
  	Source: http://www-pord.ucsd.edu/~matlab/stream.htm
  	Translated to Python by André Palóczy, January 15, 2015.
  	Modified by André Palóczy on January 15, 2015.
  	"""
  	x,y,u,v = map(np.asanyarray, (x,y,u,v))
  
  	if not x.shape==y.shape==u.shape==v.shape:
  		print("Error: Arrays (x, y, u, v) must be of equal shape.")
  		return
  
  	## Calculating grid spacings.
  	if geographic:
  		dlat, _ = np.gradient(y)
  		_, dlon = np.gradient(x)
  		deg2m = 111120.0                     # [m/deg]
  		dx = dlon*deg2m*np.cos(y*np.pi/180.) # [m]
  		dy = dlat*deg2m                      # [m]
  	else:
  		dy, _ = np.gradient(y)
  		_, dx = np.gradient(x)
  
  	ly, lx = x.shape                         # Shape of the (x,y,u,v) arrays.
  
  	## Now the main computations.
  	## Integrate velocity fields to get potential and streamfunction.
  	## Use Simpson rule summation (function CUMSIMP).
  
  	## Compute velocity potential PHI (non-rotating part).
  	if variable=='phi':
  		cx = cumsimp(u[0,:]*dx[0,:])         # Compute x-integration constant
  		cy = cumsimp(v[:,0]*dy[:,0])         # Compute y-integration constant
  		cx = np.expand_dims(cx, 0)
  		cy = np.expand_dims(cy, 1)
  		phiy = cumsimp(v*dy) + np.tile(cx, (ly,1))
  		phix = cumsimp(u.T*dx.T).T + np.tile(cy, (1,lx))
  		phi = (phix + phiy)/2.
  		return phi
  
  	## Compute streamfunction PSI (non-divergent part).
  	if variable=='psi':
  		cx = cumsimp(v[0,:]*dx[0,:])         # Compute x-integration constant
  		cy = cumsimp(u[:,0]*dy[:,0])         # Compute y-integration constant
  		cx = np.expand_dims(cx, 0)
  		cy = np.expand_dims(cy, 1)
  		psix = -cumsimp(u*dy) + np.tile(cx, (ly,1))
  		psiy = cumsimp(v.T*dx.T).T - np.tile(cy, (1,lx))
  		psi = (psix + psiy)/2.
  		return psi
  
  def cumsimp(y):
  	"""
  	F = CUMSIMP(Y)    Simpson-rule column-wise cumulative summation.
  	Numerical approximation of a function F(x) such that
  	Y(X) = dF/dX.  Each column of the input matrix Y represents
  	the value of the integrand  Y(X)  at equally spaced points
  	X = 0,1,...size(Y,1).
  	The output is a matrix  F of the same size as Y.
  	The first row of F is equal to zero and each following row
  	is the approximation of the integral of each column of matrix
  	Y up to the givem row.
  	CUMSIMP assumes continuity of each column of the function Y(X)
  	and uses Simpson rule summation.
  	Similar to the command F = CUMSUM(Y), exept for zero first
  	row and more accurate summation (under the assumption of
  	continuous integrand Y(X)).
  
  	Author: Kirill K. Pankratov, March 7, 1994.
  	Source: http://www-pord.ucsd.edu/~matlab/stream.htm
  	Translated to Python by André Palóczy, January 15, 2015.
  	"""
  	y = np.asanyarray(y)
  
  	## 3-point interpolation coefficients to midpoints.
  	## Second-order polynomial (parabolic) interpolation coefficients
  	## from  Xbasis = [0 1 2]  to  Xint = [.5 1.5]
  	c1 = 3/8.
  	c2 = 6/8.
  	c3 = -1/8.
  
  	if y.ndim==1:
  		y = np.expand_dims(y,1)
  		f = np.zeros((y.size,1))    # Initialize summation array.
  		squeeze_after = True
  	elif y.ndim==2:
  		f = np.zeros(y.shape)       # Initialize summation array.
  		squeeze_after = False
  	else:
  		print("Error: Input array has more than 2 dimensions.")
  		return
  
  	if y.size==2:                   # If only 2 elements in columns - simple average.
  		f[1,:] = (y[0,:] + y[1,:])/2.
  		return f
  	else:                           # If more than two elements in columns - Simpson summation.
  		## Interpolate values of y to all midpoints.
  		f[1:-1,:] = c1*y[:-2,:] + c2*y[1:-1,:] + c3*y[2:,:]
  		f[2:,:] = f[2:,:] + c3*y[:-2,:] + c2*y[1:-1,:] + c1*y[2:,:]
  		f[1,:] = f[1,:]*2
  		f[-1,:] = f[-1,:]*2
  
  		## Simpson (1,4,1) rule.
  		f[1:,:] = 2*f[1:,:] + y[:-1,:] + y[1:,:]
  		f = np.cumsum(f, axis=0)/6. # Cumulative sum, 6 - denominator from the Simpson rule.
  
  	if squeeze_after:
  		f = f.squeeze()
  
  	return f
  
  def rot_vec(u, v, angle=-45, degrees=True):
  	"""
  	USAGE
  	-----
  	u_rot,v_rot = rot_vec(u,v,angle=-45.,degrees=True)
  
  	Returns the rotated vector components (`u_rot`,`v_rot`)
  	from the zonal-meridional input vector components (`u`,`v`).
  	The rotation is done using the angle `angle` positive counterclockwise
  	(trigonometric convention). If `degrees` is set to `True``(default),
  	then `angle` is converted to radians.
  	is
  
  	Example
  	-------
  	>>> from matplotlib.pyplot import quiver
  	>>> from ap_tools.utils import rot_vec
  	>>> u = -1.
  	>>> v = -1.
  	>>> u2,v2 = rot_vec(u,v, angle=-30.)
  	"""
  	u,v = map(np.asanyarray, (u,v))
  	if degrees:
  		angle = angle*np.pi/180. # Degrees to radians.
  
  	u_rot = +u*np.cos(angle) + v*np.sin(angle) # Usually the across-shore component.
  	v_rot = -u*np.sin(angle) + v*np.cos(angle) # Usually the along-shore component.
  
  	return u_rot,v_rot
  
  def avgdir(dirs, degrees=False, axis=None):
      """
      USAGE
      -----
      dirm = avgdir(dirs, degrees=False, axis=None)
  
      Calculate the mean direction of an array of directions 'dirs'.
      If 'degrees' is 'False' (default), the input directions must be
      in radians. If 'degrees' is 'True', the input directions must be
      in degrees.
  
      The direction angle is measured from the ZONAL axis, i.e.,
      (0, 90, -90) deg are (Eastward, Northward, Southward).
      180 and -180 deg are both Westward.
  
      If 'axis' is 'None' (default) the mean is calculated on the
      flattened array. Otherwise, 'axis' is the index of the axis
      to calculate the mean over.
      """
      dirs = np.array(dirs)
  
      if degrees:
          dirs = dirs*np.pi/180 # Degrees to radians.
  
      uxs = np.cos(dirs)
      vys = np.sin(dirs)
      dirm = np.arctan2(vys.sum(axis=axis), uxs.sum(axis=axis))
  
      if degrees:
          dirm = dirm*180/np.pi # From radians to degrees.
  
      return dirm
  
  def lon180to360(lon):
  	"""
  	Converts longitude values in the range [-180,+180]
  	to longitude values in the range [0,360].
  	"""
  	lon = np.asanyarray(lon)
  	return (lon + 360.0) % 360.0
  
  def lon360to180(lon):
  	"""
  	Converts longitude values in the range [0,360]
  	to longitude values in the range [-180,+180].
  	"""
  	lon = np.asanyarray(lon)
  	return ((lon + 180.) % 360.) - 180.
  
  def bbox2ij(lon, lat, bbox=[-135., -85., -76., -64.], FIX_IDL=True):
      """
  	USAGE
  	-----
  	ilon_start, ilon_end, jlat_start, jlat_end = bbox2ij(lon, lat, bbox=[-135., -85., -76., -64.], FIX_IDL=True)
  
      OR
  
      (ilon_start_left, ilon_end_left, jlat_start, jlat_end), (ilon_start_right, ilon_end_right, jlat_start, jlat_end) = ...
      ... bbox2ij(lon, lat, bbox=[-135., -85., -76., -64.], FIX_IDL=True)
  
      Return indices for i,j that will completely cover the specified bounding box. 'lon' and 'lat' are 2D coordinate arrays
      (generated by meshgrid), and 'bbox' is a list like [lon_start, lon_end, lat_start, lat_end] describing the desired
      longitude-latitude box.
  
      If the specified bbox is such that it crosses the edges of the longitude array, two tuples of indices are returned.
      The first (second) tuple traces out the left (right) part of the bbox.
  
      If FIX_IDL is set to 'True' (default), the indices returned correspond to the "short route" around the globe, which
      amounts to assuming that the specified bbox crosses the International Date. If FIX_IDL is set to 'False', the
      "long route" is used instead.
  
      Example
      -------
      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> lon = np.arange(-180., 180.25, 0.25)
      >>> lat = np.arange(-90., 90.25, 0.25)
      >>> lon, lat = np.meshgrid(lon, lat)
      >>> h = np.sin(lon) + np.cos(lat)
      >>> i0, i1, j0, j1 = bbox2ij(lon, lat, bbox=[-71, -63., 39., 46])
      >>> h_subset = h[j0:j1,i0:i1]
      >>> lon_subset = lon[j0:j1,i0:i1]
      >>> lat_subset = lat[j0:j1,i0:i1]
      >>> fig, ax = plt.subplots()
      >>> ax.pcolor(lon_subset,lat_subset,h_subset)
      >>> plt.axis('tight')
  
      Original function downloaded from http://gis.stackexchange.com/questions/71630/subsetting-a-curvilinear-netcdf-file-roms-model-output-using-a-lon-lat-boundin
      Modified by André Palóczy on August 20, 2016 to handle bboxes that
      cross the International Date Line or the edges of the longitude array.
      """
      lon, lat, bbox = map(np.asanyarray, (lon, lat, bbox))
  
      # Test whether the wanted bbox crosses the International Date Line (brach cut of the longitude array).
      dlon = bbox[:2].ptp()
      IDL_BBOX=dlon>180.
      IDL_BBOX=np.logical_and(IDL_BBOX, FIX_IDL)
  
      mypath = np.array([bbox[[0,1,1,0]], bbox[[2,2,3,3]]]).T
      p = path.Path(mypath)
      points = np.vstack((lon.flatten(), lat.flatten())).T
      n, m = lon.shape
      inside = p.contains_points(points).reshape((n, m))
  
      # Fix mask if bbox goes throught the International Date Line.
      if IDL_BBOX:
          fcol=np.all(~inside, axis=0)
          flin=np.any(inside, axis=1)
          fcol, flin = map(np.expand_dims, (fcol, flin), (0, 1))
          fcol = np.tile(fcol, (n, 1))
          flin = np.tile(flin, (1, m))
          inside=np.logical_and(flin, fcol)
          print("Bbox crosses the International Date Line.")
  
      ii, jj = np.meshgrid(range(m), range(n))
      iiin, jjin = ii[inside], jj[inside]
      i0, i1, j0, j1 = min(iiin), max(iiin), min(jjin), max(jjin)
  
      SPLIT_BBOX=(i1-i0)==(m-1) # Test whether the wanted bbox crosses edges of the longitude array.
  
      # If wanted bbox crosses edges of the longitude array, return indices for the two boxes separately.
      if SPLIT_BBOX:
          Iiin = np.unique(iiin)
          ib0 = np.diff(Iiin).argmax()  # Find edge of the inner side of the left bbox.
          ib1 = ib0 + 1                 # Find edge of the inner side of the right bbox.
          Il, Ir = Iiin[ib0], Iiin[ib1] # Indices of the columns that bound the inner side of the two bboxes.
          print("Bbox crosses edges of the longitude array. Returning two sets of indices.")
          return (i0, Il, j0, j1), (Ir, i1, j0, j1)
      else:
          return i0, i1, j0, j1
  
  def xy2dist(x, y, cyclic=False, datum='WGS84'):
      """
      USAGE
      -----
      d = xy2dist(x, y, cyclic=False, datum='WGS84')
  
      Calculates a distance axis from a line defined by longitudes and latitudes
      'x' and 'y', using either the Vicenty formulae on an ellipsoidal earth
      (ellipsoid defaults to WGS84) or on a sphere (if datum=='Sphere').
  
      Example
      -------
      >>> yi, yf = -23.550520, 32.71573800
      >>> xi, xf = -46.633309, -117.161084
      >>> x, y = np.linspace(xi, xf), np.linspace(yi, yf)
      >>> d_ellipse = xy2dist(x, y, datum='WGS84')[-1]*1e-3            # [km].
      >>> d_sphere = xy2dist(x, y, datum='Sphere')[-1]*1e-3            # [km].
      >>> dd = np.abs(d_ellipse - d_sphere)
      >>> dperc = 100*dd/d_ellipse
      >>> msg = 'Difference of %.1f km over a %.0f km-long line (%.3f %% difference)'%(dd, d_ellipse, dperc)
      >>> print(msg)
      """
      if datum!="Sphere":
          xy = [LatLon(y0, x0, datum=Datums[datum]) for x0, y0 in zip(x, y)]
      else:
          xy = [LatLon_sphere(y0, x0) for x0, y0 in zip(x, y)]
      d = np.array([xy[n].distanceTo(xy[n+1]) for n in range(len(xy)-1)])
  
      return np.append(0, np.cumsum(d))
  
  def get_xtrackline(lon1, lon2, lat1, lat2, L=200, dL=10):
      """
      USAGE
      -----
      lonp, latp = get_xtrackline(lon1, lon2, lat1, lat2, L=200, dL=13)
  
      Generates a great-circle line with length 2L (with L in km) that is perpendicular to the great-circle line
      defined by the input points (lon1, lat1) and (lon2, lat2). The spacing between the points along the output
      line is dL km. Assumes a spherical Earth.
      """
      km2m = 1e3
      L, dL = L*km2m, dL*km2m
      nh = int(L/dL)
  
      p1, p2 = LatLon_sphere(lat1, lon1), LatLon_sphere(lat2, lon2)
      angperp = p1.initialBearingTo(p2) + 90
      angperpb = angperp + 180
      pm = p1.midpointTo(p2)
  
      # Create perpendicular line starting from the midpoint.
      N = range(1, nh + 1)
      pperp = []
      _ = [pperp.append(pm.destination(dL*n, angperpb)) for n in N]
      pperp.reverse()
      pperp.append(pm)
      _ = [pperp.append(pm.destination(dL*n, angperp)) for n in N]
  
      lonperp = np.array([p.lon for p in pperp])
      latperp = np.array([p.lat for p in pperp])
  
      return lonperp, latperp
  
  def get_arrdepth(arr):
      """
      USAGE
      -----
      arr_depths = get_arrdepth(arr)
  
      Determine number of nested levels in each
      element of an array of arrays of arrays...
      (or other array-like objects).
      """
      arr = np.array(arr) # Make sure first level is an array.
  
      all_nlevs = []
      for i in range(arr.size):
          nlev=0
          wrk_arr = arr[i]
          while np.size(wrk_arr)>0:
              try:
                  wrk_arr = np.array(wrk_arr[i])
              except Exception:
                  all_nlevs.append(nlev)
                  nlev=0
                  break
              nlev+=1
  
      return np.array(all_nlevs)
  
  def fpointsbox(x, y, fig, ax, nboxes=1, plot=True, pause_secs=5, return_index=True):
      """
      USAGE
      -----
      fpts = fpointsbox(x, y, fig, ax, nboxes=1, plot=True, pause_secs=5, return_index=True)
  
      Find points in a rectangle made with 2 ginput points.
      """
      fpts = np.array([])
      for n in range(nboxes):
          box = np.array(fig.ginput(n=2, timeout=0))
          try:
              xb, yb = box[:,0], box[:,1]
          except IndexError:
              print("No points selected. Skipping box \# %d."%(n+1))
              continue
          xl, xr, yd, yu = xb.min(), xb.max(), yb.min(), yb.max()
          xbox = np.array([xl, xr, xr, xl, xl])
          ybox = np.array([yd, yd, yu, yu, yd])
          fxbox, fybox = np.logical_and(x>xl, x<xr), np.logical_and(y>yd, y<yu)
          fptsi = np.logical_and(fxbox, fybox)
          if return_index:
              fptsi = np.where(fptsi)[0]
          fpts = np.append(fpts, fptsi)
          if plot:
              ax.plot(xbox, ybox, 'r', linestyle='solid', marker='o', ms=4)
              ax.plot(x[fptsi], y[fptsi], 'r', linestyle='none', marker='+', ms=5)
              plt.draw()
              fig.show()
          else:
              fig.close()
  
      if plot:
          plt.draw()
          fig.show()
          system("sleep %d"%pause_secs)
  
      return fpts
  
  def near(x, x0, npts=1, return_index=False):
      """
      USAGE
      -----
      xnear = near(x, x0, npts=1, return_index=False)
  
      Finds 'npts' points (defaults to 1) in array 'x'
      that are closest to a specified 'x0' point.
      If 'return_index' is True (defauts to False),
      then the indices of the closest points are
      returned. The indices are ordered in order of
      closeness.
      """
      x = list(x)
      xnear = []
      xidxs = []
      for n in range(npts):
          idx = np.nanargmin(np.abs(np.array(x)-x0))
          xnear.append(x.pop(idx))
          if return_index:
              xidxs.append(idx)
      if return_index: # Sort indices according to the proximity of wanted points.
          xidxs = [xidxs[i] for i in np.argsort(xnear).tolist()]
      xnear.sort()
  
      if npts==1:
          xnear = xnear[0]
          if return_index:
              xidxs = xidxs[0]
      else:
          xnear = np.array(xnear)
  
      if return_index:
          return xidxs
      else:
          return xnear
  
  def near2(x, y, x0, y0, npts=1, return_index=False):
      """
      USAGE
      -----
      xnear, ynear = near2(x, y, x0, y0, npts=1, return_index=False)
  
      Finds 'npts' points (defaults to 1) in arrays 'x' and 'y'
      that are closest to a specified '(x0, y0)' point. If
      'return_index' is True (defauts to False), then the
      indices of the closest point(s) are returned.
  
      Example
      -------
      >>> x = np.arange(0., 100., 0.25)
      >>> y = np.arange(0., 100., 0.25)
      >>> x, y = np.meshgrid(x, y)
      >>> x0, y0 = 44.1, 30.9
      >>> xn, yn = near2(x, y, x0, y0, npts=1)
      >>> print("(x0, y0) = (%f, %f)"%(x0, y0))
      >>> print("(xn, yn) = (%f, %f)"%(xn, yn))
      """
      x, y = map(np.array, (x, y))
      shp = x.shape
  
      xynear = []
      xyidxs = []
      dx = x - x0
      dy = y - y0
      dr = dx**2 + dy**2
      for n in range(npts):
          xyidx = np.unravel_index(np.nanargmin(dr), dims=shp)
          if return_index:
              xyidxs.append(xyidx)
          xyn = (x[xyidx], y[xyidx])
          xynear.append(xyn)
          dr[xyidx] = np.nan
  
      if npts==1:
          xynear = xynear[0]
          if return_index:
              xyidxs = xyidxs[0]
  
      if return_index:
          return xyidxs
      else:
          return xynear
  
  def mnear(x, y, x0, y0):
  	"""
  	USAGE
  	-----
  	xmin,ymin = mnear(x, y, x0, y0)
  
  	Finds the the point in a (lons,lats) line
  	that is closest to a specified (lon0,lat0) point.
  	"""
  	x,y,x0,y0 = map(np.asanyarray, (x,y,x0,y0))
  	point = (x0,y0)
  
  	d = np.array([])
  	for n in range(x.size):
  		xn,yn = x[n],y[n]
  		dn = distance((xn,x0),(yn,y0)) # Calculate distance point-wise.
  		d = np.append(d,dn)
  
  	idx = d.argmin()
  
  	return x[idx],y[idx]
  
  def refine(line, nref=100, close=True):
  	"""
  	USAGE
  	-----
  	ref_line = refine(line, nref=100, close=True)
  
  	Given a 1-D sequence of points 'line', returns a
  	new sequence 'ref_line', which is built by linearly
  	interpolating 'nref' points between each pair of
  	subsequent points in the original line.
  
  	If 'close' is True (default), the first value of
  	the original line is repeated at the end of the
  	refined line, as in a closed polygon.
  	"""
  	line = np.squeeze(np.asanyarray(line))
  
  	if close:
  		line = np.append(line,line[0])
  
  	ref_line = np.array([])
  	for n in range(line.shape[0]-1):
  		xi, xf = line[n], line[n+1]
  		xref = np.linspace(xi,xf,nref)
  		ref_line = np.append(ref_line, xref)
  
  	return ref_line
  
  def point_in_poly(x,y,poly):
  	"""
  	USAGE
  	-----
  	isinside = point_in_poly(x,y,poly)
  
  	Determine if a point is inside a given polygon or not
  	Polygon is a list of (x,y) pairs. This fuction
  	returns True or False.  The algorithm is called
  	'Ray Casting Method'.
  
  	Source: http://pseentertainmentcorp.com/smf/index.php?topic=545.0
  	"""
  	n = len(poly)
  	inside = False
  
  	p1x,p1y = poly[0]
  	for i in range(n+1):
  		p2x,p2y = poly[i % n]
  		if y > min(p1y,p2y):
  			if y <= max(p1y,p2y):
  				if x <= max(p1x,p2x):
  					if p1y != p2y:
  						xinters = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
  					if p1x == p2x or x <= xinters:
  						inside = not inside
  		p1x,p1y = p2x,p2y
  
  	return inside
  
  def get_mask_from_poly(xp, yp, poly, verbose=False):
  	"""
  	USAGE
  	-----
  
  	mask = get_mask_from_poly(xp, yp, poly, verbose=False)
  
  	Given two arrays 'xp' and 'yp' of (x,y) coordinates (generated by meshgrid)
  	and a polygon defined by an array of (x,y) coordinates 'poly', with
  	shape = (n,2), return a boolean array 'mask', where points that lie inside
  	'poly' are set to 'True'.
  	"""
  	print('Building the polygon mask...')
  	jmax, imax = xp.shape
  	mask = np.zeros((jmax,imax))
  	for j in range(jmax):
  		if verbose:
  			print("Row %s of %s"%(j+1,jmax))
  		for i in range(imax):
  			px, py = xp[j,i], yp[j,i]
  			# Test if this point is within the polygon.
  			mask[j,i] = point_in_poly(px, py, poly)
  
  	return mask
  
  def sphericalpolygon_area(lons, lats, R=6371000.):
  	"""
  	USAGE
  	-----
  	area = sphericalpolygon_area(lons, lats, R=6371000.)
  
  	Calculates the area of a polygon on the surface of a sphere of
  	radius R using Girard's Theorem, which states that the area of
  	a polygon of great circles is R**2 times the sum of the angles
  	between the polygons minus (N-2)*pi, where N is number of corners.
  	R = 6371000 m (6371 km, default) is a typical value for the mean
  	radius of the Earth.
  
  	Source: http://stackoverflow.com/questions/4681737/how-to-calculate-the-area-of-a-polygon-on-the-earths-surface-using-python
  	"""
  	lons, lats = map(np.asanyarray, (lons, lats))
  	N = lons.size
  
  	angles = np.empty(N)
  	for i in range(N):
  
  	    phiB1, phiA, phiB2 = np.roll(lats, i)[:3]
  	    LB1, LA, LB2 = np.roll(lons, i)[:3]
  
  	    # calculate angle with north (eastward)
  	    beta1 = greatCircleBearing(LA, phiA, LB1, phiB1)
  	    beta2 = greatCircleBearing(LA, phiA, LB2, phiB2)
  
  	    # calculate angle between the polygons and add to angle array
  	    angles[i] = np.arccos(np.cos(-beta1)*np.cos(-beta2) + np.sin(-beta1)*np.sin(-beta2))
  
  	return (np.sum(angles) - (N-2)*np.pi)*R**2
  
  def greatCircleBearing(lon1, lat1, lon2, lat2):
  	"""
  	USAGE
  	-----
  	angle = greatCircleBearing(lon1, lat1, lon2, lat2)
  
  	Calculates the angle (positive eastward) a
  	great circle passing through points (lon1,lat1)
  	and (lon2,lat2) makes with true nirth.
  
  	Source: http://stackoverflow.com/questions/4681737/how-to-calculate-the-area-of-a-polygon-on-the-earths-surface-using-python
  	"""
  	lon1, lat1, lon2, lat2 = map(np.asanyarray, (lon1, lat1, lon2, lat2))
  	dLong = lon1 - lon2
  	d2r = np.pi/180.
  
  	s = np.cos(d2r*lat2)*np.sin(d2r*dLong)
  	c = np.cos(d2r*lat1)*np.sin(d2r*lat2) - np.sin(lat1*d2r)*np.cos(d2r*lat2)*np.cos(d2r*dLong)
  
  	return np.arctan2(s, c)
  
  def weim(x, N, kind='hann', badflag=-9999, beta=14):
  	"""
  	Usage
  	-----
  	xs = weim(x, N, kind='hann', badflag=-9999, beta=14)
  
  	Description
  	-----------
  	Calculates the smoothed array 'xs' from the original array 'x' using the specified
  	window of type 'kind' and size 'N'. 'N' must be an odd number.
  
  	Parameters
  	----------
  	x       : 1D array
  	        Array to be smoothed.
  
  	N       : integer
  	        Window size. Must be odd.
  
  	kind    : string, optional
  	        One of the window types available in the numpy module:
  
  	        hann (default) : Gaussian-like. The weight decreases toward the ends. Its end-points are zeroed.
  	        hamming        : Similar to the hann window. Its end-points are not zeroed, therefore it is
  	                         discontinuous at the edges, and may produce undesired artifacts.
  	        blackman       : Similar to the hann and hamming windows, with sharper ends.
  	        bartlett       : Triangular-like. Its end-points are zeroed.
  	        kaiser         : Flexible shape. Takes the optional parameter "beta" as a shape parameter.
  	                         For beta=0, the window is rectangular. As beta increases, the window gets narrower.
  
  	        Refer to the numpy functions for details about each window type.
  
  	badflag : float, optional
  	        The bad data flag. Elements of the input array 'A' holding this value are ignored.
  
  	beta    : float, optional
  	        Shape parameter for the kaiser window. For windows other than the kaiser window,
  	        this parameter does nothing.
  
  	Returns
  	-------
  	xs      : 1D array
  	        The smoothed array.
  
  	---------------------------------------
  	André Palóczy Filho (paloczy@gmail.com)
  	June 2012
  	==============================================================================================================
  	"""
  	###########################################
  	### Checking window type and dimensions ###
  	###########################################
  	kinds = ['hann', 'hamming', 'blackman', 'bartlett', 'kaiser']
  	if ( kind not in kinds ):
  		raise ValueError('Invalid window type requested: %s'%kind)
  
  	if np.mod(N,2) == 0:
  		raise ValueError('Window size must be odd')
  
  	###########################
  	### Creating the window ###
  	###########################
  	if ( kind == 'kaiser' ): # If the window kind is kaiser (beta is required).
  		wstr = 'np.kaiser(N, beta)'
  	else: # If the window kind is hann, hamming, blackman or bartlett (beta is not required).
  		if kind == 'hann':
  			kind = 'hanning'
  
  	wstr = 'np.' + kind + '(N)'
  	w = eval(wstr)
  
  	x = np.asarray(x).flatten()
  	Fnan = np.isnan(x).flatten()
  
  	ln = (N-1)/2
  	lx = x.size
  	lf = lx - ln
  	xs = np.nan*np.ones(lx)
  
  	# Eliminating bad data from mean computation.
  	fbad=x==badflag
  	x[fbad] = np.nan
  
  	for i in range(lx):
  		if i <= ln:
  			xx = x[:ln+i+1]
  			ww = w[ln-i:]
  		elif i >= lf:
  			xx = x[i-ln:]
  			ww = w[:lf-i-1]
  		else:
  			xx = x[i-ln:i+ln+1]
  			ww = w.copy()
  
  		f = ~np.isnan(xx) # Counting only NON-NaNs, both in the input array and in the window points.
  		xx = xx[f]
  		ww = ww[f]
  
  		if f.sum() == 0: # Thou shalt not divide by zero.
  			xs[i] = x[i]
  		else:
  			xs[i] = np.sum(xx*ww)/np.sum(ww)
  
  		xs[Fnan] = np.nan # Assigning NaN to the positions holding NaNs in the input array.
  
  	return xs
  
  def smoo2(A, hei, wid, kind='hann', badflag=-9999, beta=14):
  	"""
  	Usage
  	-----
  	As = smoo2(A, hei, wid, kind='hann', badflag=-9999, beta=14)
  
  	Description
  	-----------
  	Calculates the smoothed array 'As' from the original array 'A' using the specified
  	window of type 'kind' and shape ('hei','wid').
  
  	Parameters
  	----------
  	A       : 2D array
  	        Array to be smoothed.
  
  	hei     : integer
  	        Window height. Must be odd and greater than or equal to 3.
  
  	wid     : integer
  	        Window width. Must be odd and greater than or equal to 3.
  
  	kind    : string, optional
  	        One of the window types available in the numpy module:
  
  	        hann (default) : Gaussian-like. The weight decreases toward the ends. Its end-points are zeroed.
  	        hamming        : Similar to the hann window. Its end-points are not zeroed, therefore it is
  	                         discontinuous at the edges, and may produce undesired artifacts.
  	        blackman       : Similar to the hann and hamming windows, with sharper ends.
  	        bartlett       : Triangular-like. Its end-points are zeroed.
  	        kaiser         : Flexible shape. Takes the optional parameter "beta" as a shape parameter.
  	                         For beta=0, the window is rectangular. As beta increases, the window gets narrower.
  
  	        Refer to the numpy functions for details about each window type.
  
  	badflag : float, optional
  	        The bad data flag. Elements of the input array 'A' holding this value are ignored.
  
  	beta    : float, optional
  	        Shape parameter for the kaiser window. For windows other than the kaiser window,
  	        this parameter does nothing.
  
  	Returns
  	-------
  	As      : 2D array
  	        The smoothed array.
  
  	---------------------------------------
  	André Palóczy Filho (paloczy@gmail.com)
  	April 2012
  	==============================================================================================================
  	"""
  	###########################################
  	### Checking window type and dimensions ###
  	###########################################
  	kinds = ['hann', 'hamming', 'blackman', 'bartlett', 'kaiser']
  	if ( kind not in kinds ):
  		raise ValueError('Invalid window type requested: %s'%kind)
  
  	if ( np.mod(hei,2) == 0 ) or ( np.mod(wid,2)  == 0 ):
  		raise ValueError('Window dimensions must be odd')
  
  	if (hei <= 1) or (wid <= 1):
  		raise ValueError('Window shape must be (3,3) or greater')
  
  	##############################
  	### Creating the 2D window ###
  	##############################
  	if ( kind == 'kaiser' ): # If the window kind is kaiser (beta is required).
  		wstr = 'np.outer(np.kaiser(hei, beta), np.kaiser(wid, beta))'
  	else: # If the window kind is hann, hamming, blackman or bartlett (beta is not required).
  		if kind == 'hann':
  			kind = 'hanning'
  
  		# computing outer product to make a 2D window out of the original 1d windows.
  		wstr = 'np.outer(np.' + kind + '(hei), np.' + kind + '(wid))'
  		wdw = eval(wstr)
  
  	A = np.asanyarray(A)
  	Fnan = np.isnan(A)
  	imax, jmax = A.shape
  	As = np.nan*np.ones( (imax, jmax) )
  
  	for i in range(imax):
  		for j in range(jmax):
  			### Default window parameters.
  			wupp = 0
  			wlow = hei
  			wlef = 0
  			wrig = wid
  			lh = np.floor(hei/2)
  			lw = np.floor(wid/2)
  
  			### Default array ranges (functions of the i,j indices).
  			upp = i-lh
  			low = i+lh+1
  			lef = j-lw
  			rig = j+lw+1
  
  			##################################################
  			### Tiling window and input array at the edges ###
  			##################################################
  			# Upper edge.
  			if upp < 0:
  				wupp = wupp-upp
  				upp = 0
  
  			# Left edge.
  			if lef < 0:
  				wlef = wlef-lef
  				lef = 0
  
  			# Bottom edge.
  			if low > imax:
  				ex = low-imax
  				wlow = wlow-ex
  				low = imax
  
  			# Right edge.
  			if rig > jmax:
  				ex = rig-jmax
  				wrig = wrig-ex
  				rig = jmax
  
  			###############################################
  			### Computing smoothed value at point (i,j) ###
  			###############################################
  			Ac = A[upp:low, lef:rig]
  			wdwc = wdw[wupp:wlow, wlef:wrig]
  			fnan = np.isnan(Ac)
  			Ac[fnan] = 0; wdwc[fnan] = 0 # Eliminating NaNs from mean computation.
  			fbad = Ac==badflag
  			wdwc[fbad] = 0               # Eliminating bad data from mean computation.
  			a = Ac * wdwc
  			As[i,j] = a.sum() / wdwc.sum()
  
  	As[Fnan] = np.nan # Assigning NaN to the positions holding NaNs in the input array.
  
  	return As
  
  def denan(arr):
  	"""
  	USAGE
  	-----
  	denaned_arr = denan(arr)
  
  	Remove the NaNs from an array.
  	"""
  	f = np.isnan(arr)
  	return arr[~f]
  
  def standardize(series):
  	"""
  	USAGE
  	-----
  	series2 = standardize(series)
  
  	Standardizes a series by subtracting its mean value
  	and dividing by its standard deviation. The result is
  	a dimensionless series. Inputs can be of type
  	"np.array", or "Pandas.Series"/"Pandas.TimeSeries".
  	"""
  	Mean, Std = series.mean(), series.std()
  	return (series - Mean)/Std
  
  def linear_trend(series, return_line=True):
  	"""
  	USAGE
  	-----
  	line = linear_trend(series, return_line=True)
  
  	OR
  
  	b, a, x = linear_trend(series, return_line=False)
  
  	Returns the linear fit (line = b*x + a) associated
  	with the 'series' array.
  
  	Adapted from pylab.detrend_linear.
  	"""
  	series = np.asanyarray(series)
  	x = np.arange(series.size, dtype=np.float_)
  
  	C = np.cov(x, series, bias=1) # Covariance matrix.
  	b = C[0, 1]/C[0, 0] # Angular coefficient.
  
  	a = series.mean() - b*x.mean() # Linear coefficient.
  	line = b*x + a
  
  	if return_line:
  		return line
  	else:
  		return b, a, x
  
  def thomas(A, b):
      """
      USAGE
      -----
      x = thomas(A,b)
  
      Solve Ax = b (where A is a tridiagonal matrix)
      using the Thomas Algorithm.
  
      References
      ----------
      For a step-by-step derivation of the algorithm, see
      e.g., http://www3.ul.ie/wlee/ms6021_thomas.pdf
      """
      # Step 1: Sweep rows from top to bottom,
      # calculating gammas and rhos along the way.
      N = b.size
      gam = [float(A[0,1]/A[0,0])]
      rho = [float(b[0]/A[0,0])]
      for i in range(0, N):
          rho.append(float((b[i] - A[i,i-1]*rho[-1])/(A[i,i] - A[i,i-1]*gam[-1])))
          if i<N-1: # No gamma in the last row.
              gam.append(float(A[i,i+1]/(A[i,i] - A[i,i-1]*gam[-1])))
  
      # Step 2: Substitute solutions for unknowns
      # starting from the bottom row all the way up.
      x = [] # Vector of unknowns.
      x.append(rho.pop()) # Last row is already solved.
      for i in range(N-2, -1, -1):
          x.append(float(rho.pop() - gam.pop()*x[-1]))
  
      x.reverse()
      return np.array(x)
  
  def topo_slope(lon, lat, h):
  	"""
  	USAGE
  	-----
  	lons, lats, slope = topo_slope(lon, lat, h)
  
  	Calculates bottom slope for a topography fields 'h' at
  	coordinates ('lon', 'lat') using first-order finite differences.
  	The output arrays have shape (M-1,L-1), where M,L = h.shape().
  	"""
  	lon,lat,h = map(np.asanyarray, (lon,lat,h))
  	deg2m = 1852.*60.    # m/deg.
  	deg2rad = np.pi/180. # rad/deg.
  
  	x = lon*deg2m*np.cos(lat*deg2rad)
  	y = lat*deg2m
  
  	# First-order differences, accurate to O(dx) and O(dy),
  	# respectively.
  	sx = (h[:,1:] - h[:,:-1]) / (x[:,1:] - x[:,:-1])
  	sy = (h[1:,:] - h[:-1,:]) / (y[1:,:] - y[:-1,:])
  
  	# Finding the values of the derivatives sx and sy
  	# at the same location in physical space.
  	sx = 0.5*(sx[1:,:]+sx[:-1,:])
  	sy = 0.5*(sy[:,1:]+sy[:,:-1])
  
  	# Calculating the bottom slope.
  	slope = np.sqrt(sx**2 + sy**2)
  
  	# Finding the lon,lat coordinates of the
  	# values of the derivatives sx and sy.
  	lons = 0.5*(lon[1:,:]+lon[:-1,:])
  	lats = 0.5*(lat[1:,:]+lat[:-1,:])
  	lons = 0.5*(lons[:,1:]+lons[:,:-1])
  	lats = 0.5*(lats[:,1:]+lats[:,:-1])
  
  	return lons, lats, slope
  
  def curvature_geometric(x, y):
  	"""
  	USAGE
  	-----
  	k = curvature_geometric(x, y)
  
  	Estimates the curvature k of a 2D curve (x,y) using a geometric method.
  
  	If your curve is given by two arrays, x and y, you can
  	approximate its curvature at each point by the reciprocal of the
  	radius of a circumscribing triangle with that point, the preceding
  	point, and the succeeding point as vertices. The radius of such a
  	triangle is one fourth the product of the three sides divided by its
  	area.
  
  	The curvature will be positive for curvature to the left and
  	negative for curvature to the right as you advance along the curve.
  
  	Note that if your data are too closely spaced together or subject
  	to substantial noise errors, this formula will not be very accurate.
  
  	Author: Roger Stafford
  	Source: http://www.mathworks.com/matlabcentral/newsreader/view_thread/125637
  	Translated to Python by André Palóczy, January 19, 2015.
  	"""
  	x,y = map(np.asanyarray, (x,y))
  
  	x1 = x[:-2]; x2 = x[1:-1]; x3 = x[2:]
  	y1 = y[:-2]; y2 = y[1:-1]; y3 = y[2:]
  	## a, b, and c are the three sides of the triangle.
  	a = np.sqrt((x3-x2)**2 + (y3-y2)**2)
  	b = np.sqrt((x1-x3)**2 + (y1-y3)**2)
  	c = np.sqrt((x2-x1)**2 + (y2-y1)**2)
  	## A is the area of the triangle.
  	A = 0.5*(x1*y2 + x2*y3 + x3*y1 - x1*y3 - x2*y1 - x3*y2)
  	## The reciprocal of the circumscribed radius, i.e., the curvature.
  	k = 4.0*A/(a*b*c)
  
  	return np.squeeze(k)
  
  def get_isobath(lon, lat, topo, iso, cyclic=False, smooth_isobath=False, window_length=21, win_type='barthann', **kw):
      """
      USAGE
      -----
      lon_isob, lat_isob = get_isobath(lon, lat, topo, iso, cyclic=False, smooth_isobath=False, window_length=21, win_type='barthann', **kw)
  
      Retrieves the 'lon_isob','lat_isob' coordinates of a wanted 'iso'
      isobath from a topography array 'topo', with 'lon_topo','lat_topo'
      coordinates.
      """
      lon, lat, topo = map(np.array, (lon, lat, topo))
  
      fig, ax = plt.subplots()
      cs = ax.contour(lon, lat, topo, [iso])
      coll = cs.collections[0]
      ## Test all lines to find thel ongest one.
      ## This is assumed to be the wanted isobath.
      ncoll = len(coll.get_paths())
      siz = np.array([])
      for n in range(ncoll):
          path = coll.get_paths()[n]
          siz = np.append(siz, path.vertices.shape[0])
  
      f = siz.argmax()
      xiso = coll.get_paths()[f].vertices[:, 0]
      yiso = coll.get_paths()[f].vertices[:, 1]
      plt.close()
  
      # Smooth the isobath with a moving window.
      # Periodize according to window length to avoid losing edges.
      if smooth_isobath:
          fleft = window_length//2
          fright = -window_length//2 + 1
          if cyclic:
              xl = xiso[:fleft] + 360
              xr = xiso[fright:] - 360
              yl = yiso[:fleft]
              yr = yiso[fright:]
              xiso = np.concatenate((xr, xiso, xl))
              yiso = np.concatenate((yr, yiso, yl))
              # xiso = rolling_window(xiso, window=window_length, win_type=win_type, center=True, **kw)[fleft:fright] # FIXME
              # yiso = rolling_window(yiso, window=window_length, win_type=win_type, center=True, **kw)[fleft:fright] # FIXME
          # else:
              # xiso = rolling_window(xiso, window=window_length, win_type=win_type, center=True, **kw) # FIXME
              # yiso = rolling_window(yiso, window=window_length, win_type=win_type, center=True, **kw) # FIXME
  
      return xiso, yiso
  
  def angle_isobath(lon, lat, h, isobath=100, cyclic=False, smooth_isobath=True, window_length=21, win_type='barthann', plot_map=False, **kw):
      """
      USAGE
      -----
      lon_isob, lat_isob, angle = angle_isobath(lon, lat, h, isobath=100, cyclic=False, smooth_isobath=True, window_length=21, win_type='barthann', plot_map=False, **kw)
  
      Returns the coordinates ('lon_isob', 'lat_isob') and the angle an isobath
      makes with the zonal direction for a topography array 'h' at coordinates
      ('lon', 'lat'). Defaults to the 100 m isobath.
  
      If 'smooth_isobath'==True, smooths the isobath with a rolling window of type
      'win_type' and 'window_length' points wide.
      All keyword arguments are passed to 'pandas.rolling_window()'.
  
      If 'plot_map'==True, plots a map showing
      the isobath (and its soothed version if smooth_isobath==True).
      """
      lon, lat, h = map(np.array, (lon, lat, h))
      R = 6371000.0 # Mean radius of the earth in meters (6371 km), from gsw.constants.earth_radius.
      deg2rad = np.pi/180. # [rad/deg]
  
      # Extract isobath coordinates
      xiso, yiso = get_isobath(lon, lat, h, isobath)
  
      if cyclic: # Add cyclic point.
          xiso = np.append(xiso, xiso[0])
          yiso = np.append(yiso, yiso[0])
  
      # Smooth the isobath with a moving window.
      if smooth_isobath:
          xiso = rolling_window(xiso, window=window_length, win_type=win_type, **kw)
          yiso = rolling_window(yiso, window=window_length, win_type=win_type, **kw)
  
      # From the coordinates of the isobath, find the angle it forms with the
      # zonal axis, using points k+1 and k.
      shth = yiso.size-1
      theta = np.zeros(shth)
      for k in range(shth):
          dyk = R*(yiso[k+1]-yiso[k])
          dxk = R*(xiso[k+1]-xiso[k])*np.cos(yiso[k]*deg2rad)
          theta[k] = np.arctan2(dyk,dxk)
  
      xisom = 0.5*(xiso[1:] + xiso[:-1])
      yisom = 0.5*(yiso[1:] + yiso[:-1])
  
      # Plots map showing the extracted isobath.
      if plot_map:
          fig, ax = plt.subplots()
          m = bb_map([lon.min(), lon.max()], [lat.min(), lat.max()], projection='cyl', resolution='h', ax=ax)
          m.plot(xisom, yisom, color='b', linestyle='-', zorder=3, latlon=True)
          input("Press any key to continue.")
          plt.close()
  
      return xisom, yisom, theta
  
  def isopyc_depth(z, dens0, isopyc=1027.75, dzref=1.):
      """
      USAGE
      -----
      hisopyc = isopyc_depth(z, dens0, isopyc=1027.75)
  
      Calculates the spatial distribution of the depth of a specified isopycnal 'isopyc'
      (defaults to 1027.75 kg/m3) from a 3D density array rho0 (in kg/m3) with shape
      (nz,ny,nx) and a 1D depth array 'z' (in m) with shape (nz).
  
      'dzref' is the desired resolution for the refined depth array (defaults to 1 m) which
      is generated for calculating the depth of the isopycnal. The smaller 'dzref', the smoother
      the resolution of the returned isopycnal depth array 'hisopyc'.
      """
      z, dens0 = map(np.asanyarray, (z, dens0))
      ny, nx = dens0.shape[1:]
      zref = np.arange(z.min(), z.max(), dzref)
  
      if np.ma.isMaskedArray(dens0):
          dens0 = np.ma.filled(dens0, np.nan)
  
      hisopyc = np.nan*np.ones((ny,nx))
      for j in range(ny):
          for i in range(nx):
              dens0ij = dens0[:,j,i]
              if np.logical_or(np.logical_or(isopyc<np.nanmin(dens0ij), np.nanmax(dens0ij)<isopyc), np.isnan(dens0ij).all()):
                  continue
              else:
                  dens0ref = np.interp(zref, z, dens0ij) # Refined density profile.
                  dens0refn = near(dens0ref, isopyc)
                  fz=dens0ref==dens0refn
                  try:
                      hisopyc[j,i] = zref[fz]
                  except ValueError:
                      print("Warning: More than 1 (%d) nearest depths found. Using the median of the depths for point (j=%d,i=%d)."%(fz.sum(), j, i))
                      hisopyc[j,i] = np.nanmedian(zref[fz])
  
      return hisopyc
  
  def whiten_zero(x, y, z, ax, cs, n=1, cmap=plt.cm.RdBu_r, zorder=9):
  	"""
  	USAGE
  	-----
  	whiten_zero(x, y, z, ax, cs, n=1, cmap=plt.cm.RdBu_r, zorder=9)
  
  	Changes to white the color of the 'n' (defaults to 1)
  	neighboring patches about the zero contour created
  	by a command like 'cs = ax.contourf(x, y, z)'.
  	"""
  	x, y, z = map(np.asanyarray, (x,y,z))
  	white = (1.,1.,1.)
  	cslevs = cs.levels
  	assert 0. in cslevs
  	f0=np.where(cslevs==0.)[0][0]
  	f0m, f0p = f0-n, f0+n
  	c0m, c0p = cslevs[f0m], cslevs[f0p]
  	ax.contourf(x, y, z, levels=[c0m, c0p], linestyles='none', colors=[white, white], cmap=None, zorder=zorder)
  
  def wind2stress(u, v, formula='large_pond1981-modified'):
  	"""
  	USAGE
  	-----
  	taux,tauy = wind2stress(u, v, formula='mellor2004')
  
  	Converts u,v wind vector components to taux,tauy
  	wind stress vector components.
  	"""
  	rho_air = 1.226            # kg/m3
  	mag = np.sqrt(u**2+v**2)   # m/s
  	Cd = np.zeros( mag.shape ) # Drag coefficient.
  
  	if formula=='large_pond1981-modified':
  		# Large and Pond (1981) formula
  		# modified for light winds, as
  		# in Trenberth et al. (1990).
  		f=mag<=1.
  		Cd[f] = 2.18e-3
  		f=np.logical_and(mag>1.,mag<3.)
  		Cd[f] = (0.62+1.56/mag[f])*1e-3
  		f=np.logical_and(mag>=3.,mag<10.)
  		Cd[f] = 1.14e-3
  		f=mag>=10.
  		Cd[f] = (0.49 + 0.065*mag[f])*1e-3
  	elif formula=='mellor2004':
  		Cd = 7.5e-4 + 6.7e-5*mag
  	else:
  		np.disp('Unknown formula for Cd.')
  		pass
  
  	# Computing wind stress [N/m2]
  	taux = rho_air*Cd*mag*u
  	tauy = rho_air*Cd*mag*v
  
  	return taux,tauy
  
  def gen_dates(start, end, dt='day', input_datetime=False):
  	"""
  	Returns a list of datetimes within the date range
  	from `start` to `end`, at a `dt` time interval.
  
  	`dt` can be 'second', 'minute', 'hour', 'day', 'week',
  	'month' or 'year'.
  
  	If `input_datetime` is False (default), `start` and `end`
  	must be a date in string form. If `input_datetime` is True,
  	`start` and `end` must be datetime objects.
  
  	Note
  	----
  	Modified from original function
  	by Filipe Fernandes (ocefpaf@gmail.com).
  
  	Example
  	-------
  	>>> from ap_tools.utils import gen_dates
  	>>> from datetime import datetime
  	>>> start = '1989-08-19'
  	>>> end = datetime.utcnow().strftime("%Y-%m-%d")
  	>>> gen_dates(start, end, dt='day')
  	"""
  	DT = dict(second=rrule.SECONDLY,
  		      minute=rrule.MINUTELY,
  		      hour=rrule.HOURLY,
  		      day=rrule.DAILY,
  		      week=rrule.WEEKLY,
  		      month=rrule.MONTHLY,
  		      year=rrule.YEARLY)
  
  	dt = DT[dt]
  
  	if input_datetime: # Input are datetime objects. No parsing needed.
  		dates = rrule.rrule(dt, dtstart=start, until=end)
  	else:              # Input in string form, parse into datetime objects.
  		dates = rrule.rrule(dt, dtstart=parser.parse(start), until=parser.parse(end))
  	return list(dates)
  
  def fmt_isobath(cs, fontsize=8, fmt='%g', inline=True, inline_spacing=7, manual=True, **kw):
  	"""
  	Formats the labels of isobath contours. `manual` is set to `True` by default,
  	but can be `False`, or a tuple/list of tuples with the coordinates of the labels.
  	All options are passed to plt.clabel().
  	"""
  	isobstrH = plt.clabel(cs, fontsize=fontsize, fmt=fmt, inline=inline, \
                            inline_spacing=inline_spacing, manual=manual, **kw)
  	for ih in range(0, len(isobstrH)): # Appends 'm' for meters at the end of the label.
  		isobstrh = isobstrH[ih]
  		isobstr = isobstrh.get_text()
  		isobstr = isobstr.replace('-','') + ' m'
  		isobstrh.set_text(isobstr)
  
  def float2latex(f, ndigits=1):
  	"""
  	USAGE
  	-----
  	texstr = float2latex(f, ndigits=1)
  
  	Converts a float input into a latex-formatted
  	string with 'ndigits' (defaults to 1).
  
  	Adapted from:
  	http://stackoverflow.com/questions/13490292/format-number-using-latex-notation-in-python
  	"""
  	float_str = "{0:.%se}"%ndigits
  	float_str = float_str.format(f)
  	base, exponent = float_str.split("e")
  	return "${0} \times 10^{{{1}}}$".format(base, int(exponent))
  
  def mat2npz(matname):
  	"""
  	USAGE
  	-----
  	mat2npz(matname)
  
  	Extract variables stored in a .mat file,
  	and saves them in a .npz file.
  	"""
  	d = loadmat(matname)
  	_ = d.pop('__header__')
  	_ = d.pop('__globals__')
  	_ = d.pop('__version__')
  	npzname = matname[:-4] + '.npz'
  	np.savez(npzname,**d)
  	return None
  
  def bb_map(lons, lats, ax, projection='merc', resolution='i', drawparallels=True, drawmeridians=True):
  	"""
  	USAGE
  	-----
  	m = bb_map(lons, lats, **kwargs)
  
  	Returns a Basemap instance with lon,lat bounding limits
  	inferred from the input arrays `lons`,`lats`.
  	Coastlines, countries, states, parallels and meridians
  	are drawn, and continents are filled.
  	"""
  	lons,lats = map(np.asanyarray, (lons,lats))
  	lonmin,lonmax = lons.min(),lons.max()
  	latmin,latmax = lats.min(),lats.max()
  
  	m = Basemap(llcrnrlon=lonmin,
  				urcrnrlon=lonmax,
  				llcrnrlat=latmin,
  				urcrnrlat=latmax,
  				projection=projection,
  				resolution=resolution,
  				ax=ax)
  
  	plt.ioff() # Avoid showing the figure.
  	m.fillcontinents(color='0.9', zorder=9)
  	m.drawcoastlines(zorder=10)
  	m.drawstates(zorder=10)
  	m.drawcountries(linewidth=2.0, zorder=10)
  	m.drawmapboundary(zorder=9999)
  	if drawmeridians:
  		m.drawmeridians(np.arange(np.floor(lonmin), np.ceil(lonmax), 1), linewidth=0.15, labels=[1, 0, 1, 0], zorder=12)
  	if drawparallels:
  		m.drawparallels(np.arange(np.floor(latmin), np.ceil(latmax), 1), linewidth=0.15, labels=[1, 0, 0, 0], zorder=12)
  	plt.ion()
  	return m
  
  def dots_dualcolor(x, y, z, thresh=20., color_low='b', color_high='r', marker='o', markersize=5):
  	"""
  	USAGE
  	-----
      dots_dualcolor(x, y, z, thresh=20., color_low='b', color_high='r')
  
  	Plots dots colored with a dual-color criterion,
  	separated by a threshold value.
  	"""
  	ax = plt.gca()
  	# Below-threshold dots.
  	f=z<=thresh
  	ax.plot(x[f], y[f], lw=0, marker=marker, ms=markersize, mfc=color_low, mec=color_low)
  	# Above-threshold dots.
  	f=z>thresh
  	ax.plot(x[f], y[f], lw=0, marker=marker, ms=markersize, mfc=color_high, mec=color_high)
  
  if __name__=='__main__':
    import doctest
    doctest.testmod()
  