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ocean_grid_generator.py
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ocean_grid_generator.py
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#!/usr/bin/env python
from __future__ import print_function
import numpypi.numpypi_series as np
# import numpy as np
import argparse
import sys, getopt
import datetime, os, subprocess
# Constants
PI_180 = np.pi / 180.0
# _default_Re = 6.378e6
_default_Re = 6371.0e3 # MIDAS
HUGE = 1.0e30
def chksum(x, lbl):
import hashlib
if type(x) in (float, int, np.float64):
y = np.array(x)
else:
y = np.zeros(x.shape)
y[:] = x
ymin, ymax, ymean = y.min(), y.max(), y.mean()
ysd = np.sqrt(((y - ymean) ** 2).mean())
print(hashlib.sha256(y).hexdigest(),"%10s" % lbl,
"min = %.15f" % ymin,"max = %.15f" % ymax,"mean = %.15f" % ymean,"sd = %.15f" % ysd)
def bipolar_projection(lamg, phig, lon_bp, rp, metrics_only=False):
"""Makes a stereographic bipolar projection of the input coordinate mesh (lamg,phig)
Returns the projected coordinate mesh and their metric coefficients (h^-1).
The input mesh must be a regular spherical grid capping the pole with:
latitudes between 2*arctan(rp) and 90 degrees
longitude between lon_bp and lonp+360
"""
### symmetry meridian resolution fix
phig = 90 - 2 * np.arctan(np.tan(0.5 * (90 - phig) * PI_180) / rp) / PI_180
tmp = mdist(lamg, lon_bp) * PI_180
sinla = np.sin(tmp) # This makes phis symmetric
sphig = np.sin(phig * PI_180)
alpha2 = (np.cos(tmp)) ** 2 # This makes dy symmetric
beta2_inv = (np.tan(phig * PI_180)) ** 2
rden = 1.0 / (1.0 + alpha2 * beta2_inv)
if not metrics_only:
B = sinla * np.sqrt(rden) # Actually two equations +- |B|
# Deal with beta=0
B = np.where(np.abs(beta2_inv) > HUGE, 0.0, B)
lamc = np.arcsin(B) / PI_180
##But this equation accepts 4 solutions for a given B, {l, 180-l, l+180, 360-l }
##We have to pickup the "correct" root.
##One way is simply to demand lamc to be continuous with lam on the equator phi=0
##I am sure there is a more mathematically concrete way to do this.
lamc = np.where((lamg - lon_bp > 90) & (lamg - lon_bp <= 180), 180 - lamc, lamc)
lamc = np.where((lamg - lon_bp > 180) & (lamg - lon_bp <= 270), 180 + lamc, lamc)
lamc = np.where((lamg - lon_bp > 270), 360 - lamc, lamc)
# Along symmetry meridian choose lamc
lamc = np.where((lamg - lon_bp == 90), 90, lamc) # Along symmetry meridian choose lamc=90-lon_bp
lamc = np.where((lamg - lon_bp == 270), 270, lamc) # Along symmetry meridian choose lamc=270-lon_bp
lams = lamc + lon_bp
##Project back onto the larger (true) sphere so that the projected equator shrinks to latitude \phi_P=lat0_tp
##then we have tan(\phi_s'/2)=tan(\phi_p'/2)tan(\phi_c'/2)
A = sinla * sphig
chic = np.arccos(A)
phis = 90 - 2 * np.arctan(rp * np.tan(chic / 2)) / PI_180
##Calculate the Metrics
rden2 = 1.0 / (1 + (rp * np.tan(chic / 2)) ** 2)
M_inv = rp * (1 + (np.tan(chic / 2)) ** 2) * rden2
M = 1 / M_inv
chig = (90 - phig) * PI_180
rden2 = 1.0 / (1 + (rp * np.tan(chig / 2)) ** 2)
N = rp * (1 + (np.tan(chig / 2)) ** 2) * rden2
N_inv = 1 / N
cos2phis = (np.cos(phis * PI_180)) ** 2
h_j_inv = (
cos2phis * alpha2 * (1 - alpha2) * beta2_inv * (1 + beta2_inv) * (rden ** 2)
+ M_inv * M_inv * (1 - alpha2) * rden
)
# Deal with beta=0. Prove that cos2phis/alpha2 ---> 0 when alpha, beta ---> 0
h_j_inv = np.where(np.abs(beta2_inv) > HUGE, M_inv * M_inv, h_j_inv)
h_j_inv = np.sqrt(h_j_inv) * N_inv
h_i_inv = (
cos2phis * (1 + beta2_inv) * (rden ** 2)
+ M_inv * M_inv * alpha2 * beta2_inv * rden
)
# Deal with beta=0
h_i_inv = np.where(np.abs(beta2_inv) > HUGE, M_inv * M_inv, h_i_inv)
h_i_inv = np.sqrt(h_i_inv)
if not metrics_only:
return lams, phis, h_i_inv, h_j_inv
else:
return h_i_inv, h_j_inv
def generate_bipolar_cap_mesh(Ni, Nj_ncap, lat0_bp, lon_bp, ensure_nj_even=True):
# Define a (lon,lat) coordinate mesh on the Northern hemisphere of the globe sphere
# such that the resolution of latg matches the desired resolution of the final grid along the symmetry meridian
print("Generating bipolar grid bounded at latitude ", lat0_bp)
if Nj_ncap % 2 != 0 and ensure_nj_even:
print(" Supergrid has an odd number of area cells!")
if ensure_nj_even:
print(" The number of j's is not even. Fixing this by cutting one row.")
Nj_ncap = Nj_ncap - 1
lon_g = lon_bp + np.arange(Ni + 1) * 360.0 / float(Ni)
lamg = np.tile(lon_g, (Nj_ncap + 1, 1))
latg0_cap = lat0_bp + np.arange(Nj_ncap + 1) * (90 - lat0_bp) / float(Nj_ncap)
phig = np.tile(latg0_cap.reshape((Nj_ncap + 1, 1)), (1, Ni + 1))
rp = np.tan(0.5 * (90 - lat0_bp) * PI_180)
lams, phis, h_i_inv, h_j_inv = bipolar_projection(lamg, phig, lon_bp, rp)
h_i_inv = h_i_inv[:, :-1] * 2 * np.pi / float(Ni)
h_j_inv = h_j_inv[:-1, :] * PI_180 * (90 - lat0_bp) / float(Nj_ncap)
print(" number of js=", phis.shape[0])
return lams, phis, h_i_inv, h_j_inv
def bipolar_cap_ij_array(i, j, Ni, Nj_ncap, lat0_bp, lon_bp, rp):
long = lon_bp + i * 360.0 / float(Ni)
latg = lat0_bp + j * (90 - lat0_bp) / float(Nj_ncap)
lamg = np.tile(long, (latg.shape[0], 1))
phig = np.tile(latg.reshape((latg.shape[0], 1)), (1, long.shape[0]))
h_i_inv, h_j_inv = bipolar_projection(lamg, phig, lon_bp, rp, metrics_only=True)
h_i_inv = h_i_inv * 2 * np.pi / float(Ni)
h_j_inv = h_j_inv * (90 - lat0_bp) * PI_180 / float(Nj_ncap)
return h_i_inv, h_j_inv
def bipolar_cap_metrics_quad_fast(order, nx, ny, lat0_bp, lon_bp, rp, Re=_default_Re):
print(" Calculating bipolar cap metrics via quadrature ...")
a, b = quad_positions(order)
daq = np.zeros([ny + 1, nx + 1])
dxq = np.zeros([ny + 1, nx + 1])
dyq = np.zeros([ny + 1, nx + 1])
j1d = np.empty([0])
for j in range(0, ny + 1):
j_s = b * j + a * (j + 1)
if j_s[-1] == ny:
j_s[-1] = ny - 0.001 # avoid phi=90 as this will cause errore.
# Niki:Find a way to avoid this properly.
# This could be a sign that there is still something
# wrong with the h_j_inv calculations at phi=90 (beta=0).
j1d = np.append(j1d, j_s)
i1d = np.empty([0])
for i in range(0, nx + 1):
i_s = b * i + a * (i + 1)
i1d = np.append(i1d, i_s)
# dx,dy = bipolar_cap_ij_array(i1d,j1d,nx,ny,lat0_bp,lon_bp,rp)
# Or to make it faster:
nj, ni = j1d.shape[0], i1d.shape[0] # Shape of results
dj = min(nj, max(32 * 1024 // ni, 1)) # Stride to use that fits in memory
lams, phis, dx, dy = (
np.zeros((nj, ni)),
np.zeros((nj, ni)),
np.zeros((nj, ni)),
np.zeros((nj, ni)),
)
for j in range(0, nj, dj):
je = min(nj, j + dj)
dx[j:je], dy[j:je] = bipolar_cap_ij_array(
i1d, j1d[j:je], nx, ny, lat0_bp, lon_bp, rp
)
# reshape to send for quad averaging
dx_r = dx.reshape(ny + 1, order, nx + 1, order)
dy_r = dy.reshape(ny + 1, order, nx + 1, order)
# area element
dxdy_r = dx_r * dy_r
for j in range(0, ny + 1):
for i in range(0, nx + 1):
daq[j, i] = quad_average_2d(dxdy_r[j, :, i, :])
dxq[j, i] = quad_average(dx_r[j, 0, i, :])
dyq[j, i] = quad_average(dy_r[j, :, i, 0])
daq = daq[:-1, :-1] * Re * Re
dxq = dxq[:, :-1] * Re
dyq = dyq[:-1, :] * Re
return dxq, dyq, daq
def quad_positions(n=3):
"""Returns weights wa and wb so that the element [xa,xb] is sampled at positions
x=wa(xa+xb*xb)."""
if n == 2:
return np.array([0.0, 1.0]), np.array([1.0, 0.0])
if n == 3:
return np.array([0.0, 0.5, 1.0]), np.array([1.0, 0.5, 0.0])
if n == 4:
r5 = 0.5 / np.sqrt(5.0)
return np.array([0.0, 0.5 - r5, 0.5 + r5, 1.0]), np.array([1.0, 0.5 + r5, 0.5 - r5, 0.0])
if n == 5:
r37 = 0.5 * np.sqrt(3.0 / 7.0)
return np.array([0.0, 0.5 - r37, 0.5, 0.5 + r37, 1.0]), np.array([1.0, 0.5 + r37, 0.5, 0.5 - r37, 0.0])
raise Exception("Uncoded order")
def quad_average(y):
"""Returns the average value found by quadrature at order n.
y is a list of values in order from x=-1 to x=1."""
if len(y) == 2: # 1, 1
d = 1.0 / 2.0
return d * (y[0] + y[1])
if len(y) == 3: # 1/3, 4/3, 1/3
d = 1.0 / 6.0
return d * (4.0 * y[1] + (y[0] + y[2]))
if len(y) == 4: # 1/6, 5/6, 5/6, 1/6
d = 1.0 / 12.0
return d * (5.0 * (y[1] + y[2]) + (y[0] + y[3]))
if len(y) == 5: # 9/10, 49/90, 64/90, 49/90, 9/90
d = 1.0 / 180.0
return d * (64.0 * y[2] + (49.0 * (y[1] + y[3])) + 9.0 * (y[0] + y[4]))
raise Exception("Uncoded order")
def quad_average_2d(y):
"""Returns the average value found by quadrature at order n.
y is a list of values in order from x1=-1 to x1=1 and x2=-1 to x2=1."""
if y.shape[0] != y.shape[1]:
raise Exception("Input array is not squared!")
if y.shape[0] == 2: # 1, 1
d = 1.0 / 2.0
return d * d * (y[0, 0] + y[0, 1] + y[1, 0] + y[1, 1])
if y.shape[0] == 3: # 1/3, 4/3, 1/3
d = 1.0 / 6.0
return (d*d * (y[0, 0] + y[0, 2] + y[2, 0] + y[2, 2]
+ 4.0 * (y[0, 1] + y[1, 0] + y[1, 2] + y[2, 1] + 4.0 * y[1, 1])))
if y.shape[0] == 4: # 1/6, 5/6, 5/6, 1/6
d = 1.0 / 12.0
w = np.array([1.0, 5.0, 5.0, 1.0])
ysum = 0.0
for j in range(0, y.shape[0]):
for i in range(0, y.shape[1]):
ysum = ysum + w[i] * w[j] * y[j, i]
return d * d * ysum
if y.shape[0] == 5: # 9/10, 49/90, 64/90, 49/90, 9/90
d = 1.0 / 180.0
w = np.array([9.0, 49.0, 64.0, 49.0, 9.0])
ysum = 0.0
for j in range(0, y.shape[0]):
for i in range(0, y.shape[1]):
ysum = ysum + w[i] * w[j] * y[j, i]
return d * d * ysum
raise Exception("Uncoded order")
def lagrange_interp(x, y, q):
"""Lagrange polynomial interpolation. Retruns f(q) which f(x) passes through four data
points at x[0..3], y[0..3]."""
# n - numerator, d - denominator
n0 = (q - x[1]) * (q - x[2]) * (q - x[3])
d0 = (x[0] - x[1]) * (x[0] - x[2]) * (x[0] - x[3])
n1 = (q - x[0]) * (q - x[2]) * (q - x[3])
d1 = (x[1] - x[0]) * (x[1] - x[2]) * (x[1] - x[3])
n2 = (q - x[0]) * (q - x[1]) * (q - x[3])
d2 = (x[2] - x[0]) * (x[2] - x[1]) * (x[2] - x[3])
n3 = (q - x[0]) * (q - x[1]) * (q - x[2])
d3 = (x[3] - x[0]) * (x[3] - x[1]) * (x[3] - x[2])
return ((n0 / d0) * y[0] + (n3 / d3) * y[3]) + ((n1 / d1) * y[1] + (n2 / d2) * y[2])
def lagrange_interp_6pt(x, y, q):
"""Lagrange polynomial interpolation. Retruns f(q) which f(x) passes through four data
points at x[0..5], y[0..5]."""
# n - numerator, d - denominator
n0 = (q - x[1]) * (q - x[2]) * (q - x[3]) * (q - x[4]) * (q - x[5])
d0 = (x[0] - x[1]) * (x[0] - x[2]) * (x[0] - x[3]) * (x[0] - x[4]) * (x[0] - x[5])
n1 = (q - x[0]) * (q - x[2]) * (q - x[3]) * (q - x[4]) * (q - x[5])
d1 = (x[1] - x[0]) * (x[1] - x[2]) * (x[1] - x[3]) * (x[1] - x[4]) * (x[1] - x[5])
n2 = (q - x[0]) * (q - x[1]) * (q - x[3]) * (q - x[4]) * (q - x[5])
d2 = (x[2] - x[0]) * (x[2] - x[1]) * (x[2] - x[3]) * (x[2] - x[4]) * (x[2] - x[5])
n3 = (q - x[0]) * (q - x[1]) * (q - x[2]) * (q - x[4]) * (q - x[5])
d3 = (x[3] - x[0]) * (x[3] - x[1]) * (x[3] - x[2]) * (x[3] - x[4]) * (x[3] - x[5])
n4 = (q - x[0]) * (q - x[1]) * (q - x[2]) * (q - x[3]) * (q - x[5])
d4 = (x[4] - x[0]) * (x[4] - x[1]) * (x[4] - x[2]) * (x[4] - x[3]) * (x[4] - x[5])
n5 = (q - x[0]) * (q - x[1]) * (q - x[2]) * (q - x[3]) * (q - x[4])
d5 = (x[5] - x[0]) * (x[5] - x[1]) * (x[5] - x[2]) * (x[5] - x[3]) * (x[5] - x[4])
return ((n0 / d0) * y[0] + (n5 / d5) * y[5]) + ((n1 / d1) * y[1] + (n4 / d4) * y[4]) + ((n2 / d2) * y[2] + (n3 / d3) * y[3])
def y_mercator(Ni, phi):
"""Equation (1)"""
R = Ni / (2 * np.pi)
return R * (np.log((1.0 + np.sin(phi)) / np.cos(phi)))
def phi_mercator(Ni, y):
"""Equation (2)"""
R = Ni / (2 * np.pi)
return np.arctan(np.sinh(y / R)) * (180 / np.pi) # Converted to degrees
def d_phi_mercator_dy(Ni, y):
"""Derivative of phi Equation (2) wrt y"""
R = Ni / (2 * np.pi)
return np.cosh(y/R)/(1.+(np.sinh(y / R))**2)/R
def y_mercator_rounded(Ni, phi):
y_float = y_mercator(Ni, phi)
return (np.sign(y_float) * np.round_(np.abs(y_float))).astype(int)
def generate_mercator_grid(Ni, phi_s, phi_n, lon0_M, lenlon_M, refineR,
shift_equator_to_u_point=True, ensure_nj_even=True, enhanced_equatorial=0):
print("Requesting Mercator grid with phi range: phi_s,phi_n=", phi_s, phi_n)
# Diagnose nearest integer y(phi range)
y_star = y_mercator_rounded(Ni, np.array([phi_s * PI_180, phi_n * PI_180]))
print(" y*=", y_star, "nj=", y_star[1] - y_star[0] + 1)
# Ensure that the equator (y=0) is a u-point
if y_star[0] % 2 == 0:
print(" *Equator may not be a u-point!")
# There is another check for this for the whole grid.
if shift_equator_to_u_point:
print(" *Fixing this by shifting the bounds!")
y_star[0] = y_star[0] - 1
y_star[1] = y_star[1] - 1
print(" y*=", y_star, "nj=", y_star[1] - y_star[0] + 1)
if (y_star[1] - y_star[0] + 1) % 2 == 0:
print(" *Supergrid has an odd number of area cells!")
if ensure_nj_even:
print(" *Fixing this by shifting the y_star[1] ")
y_star[1] = y_star[1] - 1
Nj = y_star[1] - y_star[0]
print(" Generating Mercator grid with phi range: phi_s,phi_n=", phi_mercator(Ni, y_star))
phi_M = phi_mercator(Ni, np.arange(y_star[0], y_star[1] + 1))
# Ensure that the equator (y=0) is included and is a u-point
equator = 0.0
equator_index = np.searchsorted(phi_M, equator)
if equator_index == 0:
raise Exception(" Ooops: Equator is not in the grid")
else:
print(" Equator is at j=", equator_index)
# Ensure that the equator (y=0) is a u-point
if equator_index % 2 == 0:
print(" *Equator is not going to be a u-point of this grid patch.")
if enhanced_equatorial:
print(" Enhancing the equator region resolution")
lagrange_interp_4pts = True
lagrange_interp_6pts = False
# Enhance the lattitude resolution between 30S and 30N
# Set a constant high res lattitude grid spanning 10 degrees centered at the Equator.
# This construction makes the whole Mercator subgrid symmetric around the Equator.
#
# MIDAS parameters. Where does this come from and how should it change with resolution?
phi_enh_d = -5.0 # Starting lattitude of enhanced resolution grid
phi_cub_d = -30 # Starting lattitude of cubic interpolation
N_cub = 132 * refineR / 2 # Number of points in the cubic interpolation for one shoulder
# MIDAS has 130, but 132 produces a result closer to 1/2 degree MIDAS grid
dphi_e = 0.13 * 2 / refineR # Enhanced resolution 10 degrees around the equator
N_enh = 40 * refineR / 2 # Number of points in the enhanced resolution below equator
if refineR == 1 and enhanced_equatorial: # Closest to SPEAR grid
phi_enh_d = -10
phi_cub_d = -20
N_cub = 29
N_enh = 55
dphi_e = -phi_enh_d / N_enh / 0.981
if refineR == 4 and enhanced_equatorial==8:
#1/8 degree refine
phi_enh_d = -10
N_enh = 2*enhanced_equatorial * abs(phi_enh_d)+1 #161
phi_cub_d = -20
N_cub = 101
dphi_e = -phi_enh_d / N_enh
lagrange_interp_4pts = False
lagrange_interp_6pts = True
if refineR == 4 and enhanced_equatorial==6:
#1/6 degree refine
phi_enh_d = -10
N_enh = 2*enhanced_equatorial * abs(phi_enh_d)+1 #121
phi_cub_d = -20
N_cub = 101 #What determines this?
dphi_e = -phi_enh_d / N_enh
j_c0d = np.where(phi_M < phi_enh_d)[0][-1] # The last index with phi_M<phi_enh_d
j_phi_cub_d = np.where(phi_M < phi_cub_d)[0][-1] # The last index with phi_M<phi_cub_d
dphi = phi_M[1:] - phi_M[0:-1]
phi1 = phi_M[0:j_phi_cub_d]
phi_e = phi_enh_d
if lagrange_interp_4pts:
nodes = [0, 1, N_cub - 2, N_cub - 1]
phi_nodes = [phi_M[j_phi_cub_d-1], phi_M[j_phi_cub_d], phi_e - dphi_e, phi_e]
q = np.arange(N_cub)
phi2 = lagrange_interp(nodes, phi_nodes, q)
elif lagrange_interp_6pts:
N_cub = 111
nodes = [0, 1, 2, N_cub - 3, N_cub - 2, N_cub - 1]
phi_nodes = [phi_M[j_phi_cub_d-1], phi_M[j_phi_cub_d], phi_M[j_phi_cub_d+1] ,
phi_e-dphi_e, phi_e, phi_e+dphi_e]
q = np.arange(N_cub)
phi2 = lagrange_interp_6pt(nodes, phi_nodes, q)
print(" Meridional range of pure Mercator=(", phi1[0], ",", phi1[-2], ") U (", -phi1[-2], ",", -phi1[0], ").")
print(" Meridional range of cubic interpolation=(", phi2[0], ",", phi2[-2], ") U (", -phi2[-2], ",",-phi2[0],").")
phi3 = np.concatenate((phi1[0:-1], phi2))
phi_s = phi3[-1]
phi4 = np.linspace(phi_s, 0, int(N_enh))
print(
" Meridional range of enhanced resolution=(", phi4[0], ",", -phi4[0], ")."
)
print(" Meridional value of enhanced resolution=", phi4[1] - phi4[0])
phi5 = np.concatenate((phi3[0:-1], phi4))
# Make the grid symmetric around the equator!!!!
phi_M = np.concatenate((phi5[0:-1], -phi5[::-1]))
# limit the upper lattitude by the requested phi_n
j_phi_n = np.where(phi_M < phi_n)[0][-1] # The last index with phi_M<phi_n
phi_M = phi_M[0:j_phi_n]
Nj = phi_M.shape[0] - 1
y_grid_M = np.tile(phi_M.reshape(Nj + 1, 1), (1, Ni + 1))
lam_M = lon0_M + np.arange(Ni + 1) * lenlon_M / float(Ni)
x_grid_M = np.tile(lam_M, (Nj + 1, 1))
# Double check is necessary for enhanced_equatorial
if y_grid_M.shape[0] % 2 == 0 and ensure_nj_even:
print(" The number of j's is not even. Fixing this by cutting one row at south.")
y_grid_M = np.delete(y_grid_M, 0, 0)
x_grid_M = np.delete(x_grid_M, 0, 0)
print(" Final Mercator grid range=", y_grid_M[0, 0], y_grid_M[-1, 0])
print(" number of js=", y_grid_M.shape[0])
return x_grid_M, y_grid_M
###
# Displaced pole cap functions
###
def displacedPoleCap_projection(lon_grid, lat_grid, z_0, r_joint):
r = np.tan((90 + lat_grid) * PI_180) / r_joint
# Find the theta that has matching resolution at the unit circle with longitude at the joint
# This is a conformal transformation of the unit circle (inverse to the one below)
e2itheta = np.cos(lon_grid * PI_180) + 1j * np.sin(lon_grid * PI_180)
e2ithetaprime = (e2itheta - z_0) / (1.0 - np.conj(z_0) * e2itheta)
# Conformal map to displace pole from r=0 to r=r_dispole
z = r * e2ithetaprime
w = (z + z_0) / (1 + np.conj(z_0) * z)
# Inverse projection from tangent plane back to sphere
lamcDP = np.angle(w, deg=True)
# lamcDP = np.arctan2(np.imag(w), np.real(w))/PI_180
# np.angle returns a value in the interval (-180,180)
# However the input grid longitude is in (-lon0,-lon0+360), e.g., (-300,60)
# We should shift the angle to be in that interval
##But we should also be careful to produce a monotonically increasing longitude, starting from lon0.
lamcDP = monotonic_bounding(lamcDP, lon_grid[0, 0])
#
rw = np.absolute(w)
phicDP = -90 + np.arctan(rw * r_joint) / PI_180
return lamcDP, phicDP
def monotonic_bounding(x, x_0):
x_im1 = x[:, 0] * 0 + x_0 # Initial value
for i in range(0, x.shape[1]):
x[:, i] = np.where(x[:, i] - x_im1[:] > 100, x[:, i] - 360, x[:, i])
x_im1[:] = x[:, i]
return x
def displacedPoleCap_baseGrid(i, j, ni, nj, lon0, lat0):
u = lon0 + i * 360.0 / float(ni)
a = -90.0
b = lat0
v = a + j * (b - a) / float(nj)
du = np.roll(u, shift=-1, axis=0) - u
dv = np.roll(v, shift=-1, axis=0) - v
return u, v, du, dv
def displacedPoleCap_mesh(i, j, ni, nj, lon0, lat0, lam_pole, r_pole, excluded_fraction=None):
long, latg, du, dv = displacedPoleCap_baseGrid(i, j, ni, nj, lon0, lat0)
lamg = np.tile(long, (latg.shape[0], 1))
phig = np.tile(latg.reshape((latg.shape[0], 1)), (1, long.shape[0]))
# Projection from center of globe to plane tangent at south pole
r_joint = np.tan((90 + lat0) * PI_180)
z_0 = r_pole * (np.cos(lam_pole * PI_180) + 1j * np.sin(lam_pole * PI_180))
lams, phis = displacedPoleCap_projection(lamg, phig, z_0, r_joint)
londp = lams[0, 0]
latdp = phis[0, 0]
if excluded_fraction is not None:
ny, nx = lamg.shape
jmin = np.ceil(excluded_fraction * ny)
jmin = jmin + np.mod(jmin, 2)
jmint = int(jmin)
return lams[jmint:, :], phis[jmint:, :], londp, latdp
else:
return lams, phis, londp, latdp
def generate_displaced_pole_grid(Ni, Nj_scap, lon0, lat0, lon_dp, r_dp):
print("Generating displaced pole grid bounded at latitude ", lat0)
print(" requested displaced pole lon,rdp=", lon_dp, r_dp)
i_s = np.arange(Ni + 1)
j_s = np.arange(Nj_scap + 1)
x, y, londp, latdp = displacedPoleCap_mesh(
i_s, j_s, Ni, Nj_scap, lon0, lat0, lon_dp, r_dp
)
print(" generated displaced pole lon,lat=", londp, latdp)
return x, y, londp, latdp
# numerical approximation of metrics coefficients h_i and h_j
def great_arc_distance(j0, i0, j1, i1, nx, ny, lon0, lat0, lon_dp, r_dp):
"""Returns great arc distance between nodes (j0,i0) and (j1,i1)"""
# https://en.wikipedia.org/wiki/Great-circle_distance
lam0, phi0, x, y = displacedPoleCap_mesh(i0, j0, nx, ny, lon0, lat0, lon_dp, r_dp)
lam1, phi1, x, y = displacedPoleCap_mesh(i1, j1, nx, ny, lon0, lat0, lon_dp, r_dp)
lam0, phi0 = lam0 * PI_180, phi0 * PI_180
lam1, phi1 = lam1 * PI_180, phi1 * PI_180
dphi, dlam = phi1 - phi0, lam1 - lam0
# Haversine formula
d = np.sin(0.5 * dphi) ** 2 + np.sin(0.5 * dlam) ** 2 * np.cos(phi0) * np.cos(phi1)
return 2.0 * np.arcsin(np.sqrt(d))
def numerical_hi(j, i, nx, ny, lon0, lat0, lon_dp, r_dp, eps, order=6):
"""Returns a numerical approximation to h_lambda"""
reps = 1.0 / eps
ds2 = great_arc_distance(j, i + eps, j, i - eps, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 2:
return 0.5 * ds2 * reps
ds4 = great_arc_distance(j, i + 2.0 * eps, j, i - 2.0 * eps, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 4:
return (8.0 * ds2 - ds4) * (1.0 / 12.0) * reps
ds6 = great_arc_distance(j, i + 3.0 * eps, j, i - 3.0 * eps, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 6:
return (45.0 * ds2 - 9.0 * ds4 + ds6) * (1.0 / 60.0) * reps
raise Exception("order not coded")
def numerical_hj(j, i, nx, ny, lon0, lat0, lon_dp, r_dp, eps, order=6):
"""Returns a numerical approximation to h_phi"""
reps = 1.0 / eps
ds2 = great_arc_distance(j + eps, i, j - eps, i, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 2:
return 0.5 * ds2 * reps
ds4 = great_arc_distance(j + 2.0 * eps, i, j - 2.0 * eps, i, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 4:
return (8.0 * ds2 - ds4) * (1.0 / 12.0) * reps
ds6 = great_arc_distance(j + 3.0 * eps, i, j - 3.0 * eps, i, nx, ny, lon0, lat0, lon_dp, r_dp)
if order == 6:
return (45.0 * ds2 - 9.0 * ds4 + ds6) * (1.0 / 60.0) * reps
raise Exception("order not coded")
def displacedPoleCap_metrics_quad(order, nx, ny, lon0, lat0, lon_dp, r_dp, Re=_default_Re):
print(" Calculating displaced pole cap metrics via quadrature ...")
a, b = quad_positions(order)
# Note that we need to include the index of the last point of the grid to do the quadrature correctly.
daq = np.zeros([ny + 1, nx + 1])
dxq = np.zeros([ny + 1, nx + 1])
dyq = np.zeros([ny + 1, nx + 1])
j1d = np.empty([0])
for j in range(0, ny + 1):
j_s = b * j + a * (j + 1)
j1d = np.append(j1d, j_s)
i1d = np.empty([0])
for i in range(0, nx + 1):
i_s = b * i + a * (i + 1)
i1d = np.append(i1d, i_s)
# numerical approximation to h_i_in and h_j_inv at quadrature points
dx = numerical_hi(j1d, i1d, nx, ny, lon0, lat0, lon_dp, r_dp, eps=1e-3, order=order)
dy = numerical_hj(j1d, i1d, nx, ny, lon0, lat0, lon_dp, r_dp, eps=1e-3, order=order)
# reshape to send for quad averaging
dx_r = dx.reshape(ny + 1, order, nx + 1, order)
dy_r = dy.reshape(ny + 1, order, nx + 1, order)
# area element
dxdy_r = dx_r * dy_r
for j in range(0, ny + 1):
for i in range(0, nx + 1):
daq[j, i] = quad_average_2d(dxdy_r[j, :, i, :])
dxq[j, i] = quad_average(dx_r[j, 0, i, :])
dyq[j, i] = quad_average(dy_r[j, :, i, 0])
daq = daq[:-1, :-1] * Re * Re
dxq = dxq[:, :-1] * Re
dyq = dyq[:-1, :] * Re
return dxq, dyq, daq
def cut_below(lam, phi, lowerlat):
nj, ni = lam.shape
for j in range(0, nj):
if phi[j, 0] > lowerlat:
break
jmin = j
# print("jmin",jmin)
return lam[jmin:, :], phi[jmin:, :]
def cut_above(lam, phi, upperlat):
nj, ni = lam.shape
for j in range(0, nj):
if phi[j, 0] > upperlat:
break
jmax = j
# print("jmax",jmax)
return lam[0:jmax, :], phi[0:jmax, :]
# utility function to plot grids
def plot_mesh_in_latlon(lam, phi, stride=1, phi_color="k", lam_color="r",
newfig=True, title=None, axis=None, block=False):
import matplotlib.pyplot as plt
# import cartopy
if phi.shape != lam.shape:
raise Exception("Ooops: lam and phi should have same shape")
nj, ni = lam.shape
if newfig:
plt.figure(figsize=(10, 10))
if axis is None:
for i in range(0, ni, stride):
plt.plot(lam[:, i], phi[:, i], lam_color)
for j in range(0, nj, stride):
plt.plot(lam[j, :], phi[j, :], phi_color)
else:
for i in range(0, ni, stride):
axis.plot(lam[:, i], phi[:, i], lam_color)#if cartopy is available add argument transform=cartopy.crs.Geodetic()
for j in range(0, nj, stride):
axis.plot(lam[j, :], phi[j, :], phi_color)#if cartopy is available add argument transform=cartopy.crs.Geodetic()
if title is not None:
plt.title(title)
if not block:
plt.show()
def plot_mesh_in_xyz(lam, phi, stride=1, phi_color="k", lam_color="r", lowerlat=None, upperlat=None,
newfig=True, title=None, axis=None, block=False):
if lowerlat is not None:
lam, phi = cut_below(lam, phi, lowerlat=lowerlat)
if upperlat is not None:
lam, phi = cut_above(lam, phi, upperlat=upperlat)
x = np.cos(phi * PI_180) * np.cos(lam * PI_180)
y = np.cos(phi * PI_180) * np.sin(lam * PI_180)
z = np.sin(phi * PI_180)
plot_mesh_in_latlon(x, y, stride=stride, phi_color=phi_color, lam_color=lam_color,
newfig=newfig, title=title, axis=None, block=False)
def displacedPoleCap_plot(x_s, y_s, lon0, lon_dp, lat0, stride=40, block=False, dplat=None):
#import cartopy.crs as ccrs
import matplotlib.pyplot as plt
plt.figure(figsize=(10, 10))
ax = plt.axes(projection="polar")
#if cartopy is available one could use the following for projection for a more accurate plot
#ccrs.NearsidePerspective(central_longitude=0.0, central_latitude=-90, satellite_height=3578400)
#ax.stock_img()
#ax.gridlines(draw_labels=True)
plot_mesh_in_latlon(x_s, y_s, stride=stride, newfig=False, axis=ax, block=block)
if dplat is not None:
ax.plot(lon_dp, dplat, color="r", marker="*") #if cartopy is available add argument transform=ccrs.Geodetic()
return ax
def mdist(x1, x2):
"""Returns positive distance modulo 360."""
return np.minimum(np.mod(x1 - x2, 360.0), np.mod(x2 - x1, 360.0))
def generate_grid_metrics_MIDAS(x, y, axis_units="degrees", Re=_default_Re, latlon_areafix=True):
nytot, nxtot = x.shape
if axis_units == "m":
metric = 1.0
if axis_units == "km":
metric = 1.0e3
if axis_units == "degrees":
metric = Re * PI_180
lv = (0.5 * (y[:, 1:] + y[:, :-1])) * PI_180
dx_i = mdist(x[:, 1:], x[:, :-1]) * PI_180
dy_i = (y[:, 1:] - y[:, :-1]) * PI_180
dx = Re * np.sqrt(dy_i ** 2 + (dx_i * np.cos(lv)) ** 2)
lu = (0.5 * (y[1:, :] + y[:-1, :])) * PI_180
dx_j = mdist(x[1:, :], x[:-1, :]) * PI_180
dy_j = (y[1:, :] - y[:-1, :]) * PI_180
dy = Re * np.sqrt(dy_j ** 2 + (dx_j * np.cos(lu)) ** 2)
ymid_j = 0.5 * (y + np.roll(y, shift=-1, axis=0))
ymid_i = 0.5 * (y + np.roll(y, shift=-1, axis=1))
dy_j = np.roll(y, shift=-1, axis=0) - y
dy_i = np.roll(y, shift=-1, axis=1) - y
dx_i = mdist(np.roll(x, shift=-1, axis=1), x)
dx_j = mdist(np.roll(x, shift=-1, axis=0), x)
if latlon_areafix:
sl = np.sin(lv)
dx_i = mdist(x[:, 1:], x[:, :-1]) * PI_180
area = (Re ** 2) * ((0.5 * (dx_i[1:, :] + dx_i[:-1, :])) * (sl[1:, :] - sl[:-1, :]))
else:
area = 0.25 * ((dx[1:, :] + dx[:-1, :]) * (dy[:, 1:] + dy[:, :-1]))
return dx, dy, area
def angle_x(x, y):
"""Returns the orientation angle of the grid box"""
if x.shape != y.shape:
raise Exception("Input arrays do not have the same shape!")
angle_dx = np.zeros(x.shape)
# The corrected version of angle_dx, in addition to including spherical metrics, is centered in the interior and one-sided at the grid edges
angle_dx[:, 1:-1] = np.arctan2(y[:, 2:] - y[:, :-2], (x[:, 2:] - x[:, :-2]) * np.cos(y[:, 1:-1] * PI_180))
angle_dx[:, 0] = np.arctan2(y[:, 1] - y[:, 0], (x[:, 1] - x[:, 0]) * np.cos(y[:, 0] * PI_180))
angle_dx[:, -1] = np.arctan2(y[:, -1] - y[:, -2], (x[:, -1] - x[:, -2]) * np.cos(y[:, -1] * PI_180))
angle_dx = angle_dx / PI_180
return angle_dx
def metrics_error(
dx_,
dy_,
area_,
Ni,
lat1,
lat2=90,
Re=_default_Re,
bipolar=False,
displaced_pole=-999,
excluded_fraction=None,
):
exact_area = (2 * np.pi * (Re ** 2) * np.abs(np.sin(lat2 * PI_180) - np.sin(lat1 * PI_180)))
exact_lat_arc_length = np.abs(lat2 - lat1) * PI_180 * Re
exact_lon_arc_length = np.cos(lat1 * PI_180) * 2 * np.pi * Re
grid_lat_arc_length = np.sum(dy_[:, Ni // 4])
grid_lon_arc_length = np.sum(dx_[0, :])
if lat1 > lat2:
grid_lon_arc_length = np.sum(dx_[-1, :])
if bipolar:
# length of the fold
grid_lon_arc_length2 = np.sum(dx_[-1, :])
# This must be 4*grid_lat_arc_length
lon_arc2_error = 100* (grid_lon_arc_length2 / 4 - exact_lat_arc_length)/ exact_lat_arc_length
area_error = 100 * (np.sum(area_) - exact_area) / exact_area
lat_arc_error = 100 * (grid_lat_arc_length - exact_lat_arc_length) / exact_lat_arc_length
lon_arc_error = 100 * (grid_lon_arc_length - exact_lon_arc_length) / exact_lon_arc_length
if displaced_pole != -999:
antipole = displaced_pole + Ni // 2
if displaced_pole > Ni // 2:
antipole = displaced_pole - Ni // 2
grid_lat_arc_length = np.sum(dy_[:, displaced_pole]) + np.sum(dy_[:, antipole])
lat_arc_error = 100 * (grid_lat_arc_length - 2.0 * exact_lat_arc_length)/ exact_lat_arc_length
if excluded_fraction:
print(" Cannot estimate area and dy accuracies with excluded_fraction (doughnut)! ")
if bipolar:
return area_error, lat_arc_error, lon_arc_error, lon_arc2_error
else:
return area_error, lat_arc_error, lon_arc_error
def write_nc(x,y,dx,dy,area,angle_dx,axis_units="degrees",fnam=None,format="NETCDF3_64BIT",
description=None,history=None,source=None,no_changing_meta=None,debug=False):
import netCDF4 as nc
if fnam is None:
fnam = "supergrid.nc"
fout = nc.Dataset(fnam, "w", clobber=True, format=format)
if debug:
chksum(x, "x")
chksum(y, "y")
chksum(dx, "dx")
chksum(dy, "dy")
chksum(area, "area")
chksum(angle_dx, "angle_dx")
ny = area.shape[0]
nx = area.shape[1]
nyp = ny + 1
nxp = nx + 1
print(" Writing netcdf file with ny,nx= ", ny, nx)
nyp = fout.createDimension("nyp", nyp)
nxp = fout.createDimension("nxp", nxp)
ny = fout.createDimension("ny", ny)
nx = fout.createDimension("nx", nx)
string = fout.createDimension("string", 255)
tile = fout.createVariable("tile", "S1", ("string"))
yv = fout.createVariable("y", "f8", ("nyp", "nxp"))
xv = fout.createVariable("x", "f8", ("nyp", "nxp"))
yv.units = "degrees"
xv.units = "degrees"
yv[:] = y
xv[:] = x
stringvals = np.empty(1, "S" + repr(len(tile)))
stringvals[0] = "tile1"
tile[:] = nc.stringtochar(stringvals)
dyv = fout.createVariable("dy", "f8", ("ny", "nxp"))
dyv.units = "meters"
dyv[:] = dy
dxv = fout.createVariable("dx", "f8", ("nyp", "nx"))
dxv.units = "meters"
dxv[:] = dx
areav = fout.createVariable("area", "f8", ("ny", "nx"))
areav.units = "m2"
areav[:] = area
anglev = fout.createVariable("angle_dx", "f8", ("nyp", "nxp"))
anglev.units = "degrees"
anglev[:] = angle_dx
# global attributes
if not no_changing_meta:
fout.history = history
fout.description = description
fout.source = source
fout.sync()
fout.close()
def generate_latlon_grid(lni, lnj, llon0, llen_lon, llat0, llen_lat, ensure_nj_even=True):
print("Generating regular lat-lon grid between latitudes ", llat0, llat0 + llen_lat)
llonSP = llon0 + np.arange(lni + 1) * llen_lon / float(lni)
llatSP = llat0 + np.arange(lnj + 1) * llen_lat / float(lnj)
if llatSP.shape[0] % 2 == 0 and ensure_nj_even:
print(" The number of j's is not even. Fixing this by cutting one row at south.")
llatSP = np.delete(llatSP, 0, 0)
llamSP = np.tile(llonSP, (llatSP.shape[0], 1))
lphiSP = np.tile(llatSP.reshape((llatSP.shape[0], 1)), (1, llonSP.shape[0]))
print(" generated regular lat-lon grid between latitudes ", lphiSP[0, 0], lphiSP[-1, 0])
print(" number of js=", lphiSP.shape[0])
return llamSP, lphiSP
def usage():
print(
"ocean_grid_generator.py -f <output_grid_filename> -r <inverse_degrees_resolution> [--rdp=<displacement_factor/0.2> --exfracdp=0.5 --south_cutoff_ang=<degrees_south_to_start> --south_cutoff_row=<rows_south_to_cut> --match_dy bp so --even_j --plot --write_subgrid_files --enhanced_equatorial --no-metrics --grids=sc]"
)
def main(
inverse_resolution,
gridfilename="ocean_hgrid.nc",
r_dp=0.0,
lon_dp=80.0,
lat_dp=-99.0,
exfracdp=None,
south_cutoff_row=0,
south_cutoff_ang=-90.0,
reproduce_MIDAS_grids=False,
write_subgrid_files=False,
plotem=False,
no_changing_meta=False,
enhanced_equatorial=0,
debug=False,
grids=["bipolar", "mercator", "so", " sc", "all"],
match_dy=["bp", "so", " sc", "p125sc", ""],
skip_metrics=False,
ensure_nj_even=False,
shift_equator_to_u_point=True,
bipolar_lower_lat=-99.0,
mercator_lower_lat=-99.0,
mercator_upper_lat=-99.0,
south_ocean_lower_lat=-99.0,
south_ocean_upper_lat=-99.0,
no_south_cap=False,
):
known_options=["bp", "so", "p125sc", ""]
unknown=list(set(match_dy).difference(known_options))
if len(unknown) != 0:
print("Unknown options in match_dy: ",unknown)
print("Known options are one or more of ",known_options)
sys.exit(2)
# fraction of dp grid to be excluded/cut, this particular value was used for the OM4 1/4 degree grid
doughnut = 0.28 * 7 / 4
doughnut = exfracdp if (exfracdp is not None) else doughnut
south_cap = not no_south_cap
calculate_metrics = not skip_metrics
degree_resolution_inverse = inverse_resolution
hasBP = False
hasMerc = False
hasSO = False
hasSC = False
# Exit if mutually exclusive arguments are provided
if r_dp != 0.0 and lat_dp > -90.0:
print("Cannot specify both --rdp and --latdp for the displaced pole!")
usage()
sys.exit(2)
# Information to write in file as metadata
if not no_changing_meta:
import socket
host = str(socket.gethostname())
scriptpath = sys.argv[0]
scriptbasename = (subprocess.check_output("basename " + scriptpath, shell=True).decode("ascii").rstrip("\n"))
scriptdirname = (
subprocess.check_output("dirname " + scriptpath, shell=True)
.decode("ascii")
.rstrip("\n")
)
scriptgithash = (
subprocess.check_output(
"cd " + scriptdirname + ";git rev-parse HEAD; exit 0",
stderr=subprocess.STDOUT,
shell=True,
)
.decode("ascii")
.rstrip("\n")
)
scriptgitMod = (
subprocess.check_output(
"cd "
+ scriptdirname
+ ";git status --porcelain "
+ scriptbasename
+ " | awk '{print $1}' ; exit 0",
stderr=subprocess.STDOUT,
shell=True,
)
.decode("ascii")
.rstrip("\n")
)
if "M" in str(scriptgitMod):
scriptgitMod = " , But was localy Modified!"
hist = "This grid file was generated via command " + " ".join(sys.argv)
if not no_changing_meta:
hist = hist + " on " + str(datetime.date.today()) + " on platform " + host
desc = (
"This is an orthogonal coordinate grid for the Earth with a nominal resoution of "
+ str(1 / degree_resolution_inverse)
+ " degrees along the equator. "
)
source = ""
if not no_changing_meta:
source = source + scriptpath + " had git hash " + scriptgithash + scriptgitMod
source = (
source
+ ". To obtain the grid generating code do: git clone https://github.com/nikizadehgfdl/grid_generation.git ; cd grid_generation; git checkout "
+ scriptgithash
)
import time
start_time = time.time()
# Specify the default grid properties
refineS = 2 # factor 2 is for supergrid
refineR = degree_resolution_inverse
lenlon = 360 # global longitude range
lon0 = -300.0 # Starting longitude of the map
Ni = int(refineR * refineS * lenlon) #Nominal zonal resolution at the equator
###
###Mercator grid
###
#We want to generate grids close to the existing OM4p25 and OM4p5 produced by MIDAS
# MIDAS OM4p25 grid shape:(2160, 2881),
# j-extents: (SC60 0:59) (SO220 60:279) (MERC1400 280:1679) (BPC481 1680:2160)
# lat extents: (-80.43,-78.04),(-78.0,-66.91),(-66.86,64.0040), (64.06,90.0)
# MIDAS has nominal starting latitude for Mercator grid = -65 for 1/4 degree, -70 for 1/2 degree
# MIDAS has nominal latitude range of Mercator grid = 125 for 1/4 degree, 135 for 1/2 degree
#
# The following choice results the same number of points as in the original MIDA-OM4p25 (sub)grids:
phi_s_Merc, phi_n_Merc = -66.85954725, 64.05895973
#Optionally override from arguments
if mercator_upper_lat > -90:
phi_n_Merc = mercator_upper_lat
if mercator_lower_lat > -90:
phi_s_Merc = mercator_lower_lat
#Overrides to generate grids close to existing grids which were generated by MIDAS
if refineR == 2: #Closest to OM4p5 grid
#These give a 1/2 degree enhanced equatorial very close to MIDAS result
phi_s_Merc, phi_n_Merc = -68.0, 65.0
if refineR == 1 and enhanced_equatorial: #Closest to SPEAR grid
# shift_equator_to_u_point=True