Commit 8d72a548 authored by Eva Börgens's avatar Eva Börgens
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remove typos from docstring and export Docstrings as documentation

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<header>
<h1 class="title">Module <code>src.covariance</code></h1>
</header>
<section id="section-intro">
<p>Module handling the computation of the covariances and uncertainties</p>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">&#34;&#34;&#34;
Module handling the computation of the covariances and uncertainties
&#34;&#34;&#34;
import math
from typing import Dict
import scipy.special as sp
import numpy as np
rho = np.pi/180
R = 6367.4447
def get_grid_area(lon : np.ndarray, lat : np.ndarray) -&gt; np.ndarray:
&#34;&#34;&#34;
Function getting the area weights of a coordinate list
Args:
lon : np.ndarray
array of longitudes
lat : np.ndarray
array of latitudes
Returns:
np.ndarray
area per grid point
&#34;&#34;&#34;
area = np.zeros(len(lat))
delta_lon = np.abs(np.unique(lon)[1]-np.unique(lon)[0])
delta_lat = np.abs(np.unique(lat)[1] - np.unique(lat)[0])
for i, lat_i in enumerate(lat):
area[i] = get_area(0, lat_i-delta_lat/2, delta_lon, lat_i+delta_lat/2)
return area
def get_area(lon_ll: float, lat_ll: float, lon_ur: float,lat_ur: float) -&gt; float:
&#34;&#34;&#34;
returns area of rectangular on sphere, defined by corner points lower left ll and upper right ur
Args:
lon_ll: float
lon of lower left corner
lat_ll: float
lat of lower left corner
lon_ur: float
lon of upper right corner
lat_ur: float
lat of upper right corner
Returns:
float
area of grid point
&#34;&#34;&#34;
area = np.abs((lon_ll-lon_ur)*rho)*np.abs((np.sin(lat_ll*rho)-np.sin(lat_ur*rho)))*R**2
return area
def cov_function_lat(lat1: float, lat2: float,
dist: float,
theta: float,
ny: float,
a_0: float, k_a_0_2: float, k_a_0_4: float, k_a_0_6: float, k_a_0_8: float,
a_1: float, k_a_1_2: float, k_a_1_4: float, k_a_1_6: float, k_a_1_8: float,
c_0: float, k_2: float, k_4: float, k_6: float, k_8: float) -&gt; float:
&#34;&#34;&#34;
Function to compute covariance function between two points according to
publication Boergens et al. 2020
Args:
lat1, lat2: float
Latitude of the two points
d: float
distance between points
theta: float
azimuth angle between points
ny: float
order of Bessle function
a0: float
anisotropy parameter
ka0_2, ka0_4, ka0_6, ka0_8: float
Legende polynome Args for a0
a1: float
isotropy shape parameter
ka1_2, ka1_4, ka1_6, ka1_8: float
Legende polynome Args for a1
c0: float
amplitude parameter
k2,k4, k6, k8: float
Legende polynome Args for c0
Returns:
float
Covariance
&#34;&#34;&#34;
k = [1, 0, k_2, 0, k_4, 0, k_6, 0, k_8]
P_1 = sum_legendre(len(k) - 1, k, lat1 * rho)
P_2 = sum_legendre(len(k) - 1, k, lat2 * rho)
k_a_0 = [1, 0, k_a_0_2, 0, k_a_0_4, 0, k_a_0_6, 0, k_a_0_8]
P_a_0_1 = sum_legendre(len(k_a_0) - 1, k_a_0, np.mean([lat1, lat2]) * rho)
A_0 = P_a_0_1 * a_0
k_a_1 = [1, 0, k_a_1_2, 0, k_a_1_4, 0, k_a_1_6, 0, k_a_1_8]
P_a_1_1 = sum_legendre(len(k_a_1) - 1, k_a_1, np.mean([lat1, lat2]) * rho)
A_1 = P_a_1_1 * a_1
if dist==0:
dist = 0.001
C_0 = yaglom(dist, theta, A_0, A_1, c_0, ny)
Cov = P_1*P_2*C_0
return Cov
def legendre_polynome(n: int, lat: float) -&gt; float:
&#34;&#34;&#34;
Computes Legendre Polynome of degree n at given latitude lat
Args:
n: int
lat: float
Returns:
float
&#34;&#34;&#34;
sin_lat = np.sin(lat)
P_n = sp.legendre(n)
P_n_x = P_n(sin_lat)
return P_n_x
def sum_legendre(n_max: int, leg_weights: np.ndarray, lat:float) -&gt; float:
&#34;&#34;&#34;
Computes weighted sum of Legendre Polynomes
Args:
n_max: int
maximum degree of Legendre Polynomes
leg_weights: np.ndarray
arrays of the weights for the sum
lat: float
latitude where the Legendre Polynomes are summed
Returns:
float
&#34;&#34;&#34;
if n_max!=len(leg_weights)-1:
raise(&#39;N_max coefficients are needed &#39;)
p_sum = 0
for i in range(n_max+1):
p_sum += leg_weights[i] * legendre_polynome(i, lat)
return p_sum
def yaglom(dist: float,
theta: float,
a_0: float,
a_1: float,
c_0: float,
ny: float) -&gt; float:
&#34;&#34;&#34;
Function to compute the adapted Yaglom function
Args:
dist: float
spherical distance
theta: float
azimut angel
a_0: float
anisotropic width parameter
a_1: float
isotropic width parameter
c_0: float
global scaling factor
ny: int
Order of Bessel function
Returns:
float
&#34;&#34;&#34;
alpha = a_0 * np.sin(theta) + a_1
cov = c_0 / (alpha * dist) ** ny * sp.jv(ny, alpha * dist)
return cov
def distance(lon_0: float, lat_0: float, lon_1: float, lat_1: float):
&#34;&#34;&#34;
convert geograpic coordinates to spherical distances
Args:
lon_0: float
[degree]
lat_0: float
[degree]
lon_1: float
[degree]
lat_1: float
[degree]
Returns:
float
&#34;&#34;&#34;
lon_1_rad = lon_1 * rho
lat_1_rad = lat_1 * rho
lon_0_rad = lon_0 * rho
lat_0_rad = lat_0 * rho
sin_lat0 = math.sin(lat_0_rad)
cos_lat0 = math.cos(lat_0_rad)
cos_lon = math.cos(lon_1_rad-lon_0_rad)
sin_lat = math.sin(lat_1_rad)
cos_lat = math.cos(lat_1_rad)
hilf = sin_lat*sin_lat0 + cos_lat*cos_lat0*cos_lon
d_i = R * math.acos(hilf)
if lon_0==lon_1 and lat_0==lat_1:
d_i =0
return d_i
def azimut_angle(lon_0: float, lat_0: float, lon_1: float, lat_1: float):
&#34;&#34;&#34;
get azimut angle between geograpic coordinates
Args:
lon_0: float
[degree]
lat_0: float
[degree]
lon_1: float
[degree]
lat_1: float
[degree]
Returns:
float
&#34;&#34;&#34;
lat_1_rad = lat_1 * rho
lat_0_rad = lat_0 * rho
lon_0_rad = lon_0 * rho
lon_1_rad = lon_1 * rho
sin_lat0 = math.sin(lat_0_rad)
cos_lat0 = math.cos(lat_0_rad)
tan_lat = math.tan(lat_1_rad)
cos_lon = math.cos(lon_1_rad-lon_0_rad)
sin_lon = math.sin(lon_1_rad-lon_0_rad)
alpha = math.atan2(sin_lon,(cos_lat0*tan_lat-sin_lat0*cos_lon))
if lat_1 == 90:
alpha=0
elif lat_1 ==-90:
alpha = np.pi
return alpha
def compute_covariance(region_coords: np.ndarray,
gridstd: np.ndarray,
flag_uncertainty: bool, flag_matrix: bool) -&gt; Dict[str, np.ndarray]:
&#34;&#34;&#34;
Function to compute the covariances for a region
Args:
region_coords: np.ndarray
coordinates of region, size [n,2]
gridstd: np.ndarray
standard deviation for each grid point, size[n]
flag_uncertainty: bool
flag, return uncertainty of mean tws of region
flag_matrix: bool
flag, return covariance matrix of region
Returns:
Dict[str, np.ndarray]
&#34;&#34;&#34;
lon = np.array([r[0] for r in region_coords])
lat = np.array([r[1] for r in region_coords])
n = len(lat)
m = len(gridstd)
# Grid weights
area = get_grid_area(lon, lat)
region_size = np.sum(area)
# Parameter of covariance function
x = np.array([1.09e-03, -2.92, -1.90, -0.86, -4.60,
2.66e-03, 0.40, -0.30, 0.07, 0.32,
3.74, -0.2, -0.27, -0.17, -0.22])
if flag_uncertainty:
var_model = np.zeros(m)
else:
var_model = None
if flag_matrix:
cov_model = np.zeros((m,n,n))
else:
cov_model = None
for ii in range(n):
dist = 0
theta = np.pi * 2
corr = cov_function_lat(lat[ii], lat[ii], dist, theta, 2, *x)
w_1 = area[ii] / region_size
w_2 = area[ii] / region_size
for j in range(m):
sigma = gridstd[j]
if np.isnan(sigma):
continue
if flag_uncertainty:
var_model[j] += w_1 * w_2 * sigma **2 * corr
if flag_matrix:
cov_model[j,ii,ii] = sigma **2 * corr
for jj in range(ii+1, n):
dist = distance(lon[ii], lat[ii], lon[jj], lat[jj])
theta = azimut_angle(lon[ii], lat[ii], lon[jj], lat[jj])
corr = cov_function_lat(lat[ii], lat[jj], dist, theta, 2, *x)
w_1 = area[ii] / region_size
w_2 = area[jj] / region_size
for j in range(m):
sigma = gridstd[j]
if np.isnan(sigma):
continue
if flag_uncertainty:
var_model[j] += w_1 * w_2 * sigma**2 * corr
if flag_matrix:
cov_model[j, ii, jj] = sigma ** 2 * corr
cov_model[j, jj, ii] = sigma ** 2 * corr
result = {}
if flag_uncertainty:
std_model = np.sqrt(var_model)
result[&#39;uncertainty&#39;] = std_model
if flag_matrix:
result[&#39;matrix&#39;] = cov_model
return result
def get_timeseries(grid, lon_grid, lat_grid, region_coords):
&#34;&#34;&#34;
Returns mean tws time series of region
Args:
grid: np.ndarray
tws grid, size [t,n,m]
lon_grid: np.ndarray
longitude of grid, size [m]
lat_grid: np.ndarray
latitude of grid, size [n]
region_coords: np.ndarray
coordinates of region of interest, size [l,2]
Returns:
np.ndarray
size [t]
&#34;&#34;&#34;
lon_region = np.array([r[0] for r in region_coords])
lat_region = np.array([r[1] for r in region_coords])
if len(np.isin(lon_grid, lon_region).nonzero()[0])&lt; len(np.unique(lon_region)) or \
len(np.isin(lat_grid, lat_region).nonzero()[0])&lt; len(np.unique(lat_region)):
print(&#34;Warning: Region coordinates are not located on the given TWS grid, thus no &#34;
&#34;mean tws time series can be computed. Continue without timeseries output&#34;)
return None, False
# Grid weights
area = get_grid_area(lon_region, lat_region)
region_id_lat = np.array([np.nonzero(lat_grid==l)[0] for l in lat_region]).squeeze()
region_id_lon = np.array([np.nonzero(lon_grid==l)[0] for l in lon_region]).squeeze()
timeseries = np.zeros(grid.shape[0])
for i in range(grid.shape[0]):
grid_part = grid[i,:,:]
timeseries[i] = np.nansum(area * grid_part[region_id_lat, region_id_lon]) / np.sum(area)
return timeseries, True</code></pre>
</details>
</section>
<section>
</section>
<section>
</section>
<section>
<h2 class="section-title" id="header-functions">Functions</h2>
<dl>
<dt id="src.covariance.azimut_angle"><code class="name flex">
<span>def <span class="ident">azimut_angle</span></span>(<span>lon_0: float, lat_0: float, lon_1: float, lat_1: float)</span>
</code></dt>
<dd>
<div class="desc"><p>get azimut angle between geograpic coordinates</p>
<p>Args:
lon_0: float
[degree]
lat_0: float
[degree]
lon_1: float
[degree]
lat_1: float
[degree]</p>
<h2 id="returns">Returns</h2>
<p>float</p></div>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">def azimut_angle(lon_0: float, lat_0: float, lon_1: float, lat_1: float):
&#34;&#34;&#34;
get azimut angle between geograpic coordinates
Args:
lon_0: float
[degree]
lat_0: float
[degree]
lon_1: float
[degree]
lat_1: float
[degree]
Returns:
float
&#34;&#34;&#34;
lat_1_rad = lat_1 * rho
lat_0_rad = lat_0 * rho
lon_0_rad = lon_0 * rho
lon_1_rad = lon_1 * rho
sin_lat0 = math.sin(lat_0_rad)
cos_lat0 = math.cos(lat_0_rad)
tan_lat = math.tan(lat_1_rad)
cos_lon = math.cos(lon_1_rad-lon_0_rad)
sin_lon = math.sin(lon_1_rad-lon_0_rad)
alpha = math.atan2(sin_lon,(cos_lat0*tan_lat-sin_lat0*cos_lon))
if lat_1 == 90:
alpha=0
elif lat_1 ==-90:
alpha = np.pi
return alpha</code></pre>
</details>
</dd>
<dt id="src.covariance.compute_covariance"><code class="name flex">
<span>def <span class="ident">compute_covariance</span></span>(<span>region_coords: numpy.ndarray, gridstd: numpy.ndarray, flag_uncertainty: bool, flag_matrix: bool) ‑> Dict[str, numpy.ndarray]</span>
</code></dt>
<dd>
<div class="desc"><p>Function to compute the covariances for a region</p>
<p>Args:
region_coords: np.ndarray
coordinates of region, size [n,2]
gridstd: np.ndarray
standard deviation for each grid point, size[n]
flag_uncertainty: bool
flag, return uncertainty of mean tws of region
flag_matrix: bool
flag, return covariance matrix of region</p>
<h2 id="returns">Returns</h2>
<p>Dict[str, np.ndarray]</p></div>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">def compute_covariance(region_coords: np.ndarray,
gridstd: np.ndarray,
flag_uncertainty: bool, flag_matrix: bool) -&gt; Dict[str, np.ndarray]:
&#34;&#34;&#34;
Function to compute the covariances for a region
Args:
region_coords: np.ndarray
coordinates of region, size [n,2]
gridstd: np.ndarray
standard deviation for each grid point, size[n]
flag_uncertainty: bool
flag, return uncertainty of mean tws of region
flag_matrix: bool
flag, return covariance matrix of region
Returns:
Dict[str, np.ndarray]
&#34;&#34;&#34;
lon = np.array([r[0] for r in region_coords])
lat = np.array([r[1] for r in region_coords])
n = len(lat)
m = len(gridstd)
# Grid weights
area = get_grid_area(lon, lat)
region_size = np.sum(area)
# Parameter of covariance function
x = np.array([1.09e-03, -2.92, -1.90, -0.86, -4.60,
2.66e-03, 0.40, -0.30, 0.07, 0.32,
3.74, -0.2, -0.27, -0.17, -0.22])
if flag_uncertainty:
var_model = np.zeros(m)
else:
var_model = None
if flag_matrix:
cov_model = np.zeros((m,n,n))
else:
cov_model = None
for ii in range(n):
dist = 0
theta = np.pi * 2
corr = cov_function_lat(lat[ii], lat[ii], dist, theta, 2, *x)
w_1 = area[ii] / region_size
w_2 = area[ii] / region_size
for j in range(m):
sigma = gridstd[j]
if np.isnan(sigma):
continue
if flag_uncertainty:
var_model[j] += w_1 * w_2 * sigma **2 * corr
if flag_matrix:
cov_model[j,ii,ii] = sigma **2 * corr
for jj in range(ii+1, n):
dist = distance(lon[ii], lat[ii], lon[jj], lat[jj])
theta = azimut_angle(lon[ii], lat[ii], lon[jj], lat[jj])
corr = cov_function_lat(lat[ii], lat[jj], dist, theta, 2, *x)
w_1 = area[ii] / region_size
w_2 = area[jj] / region_size
for j in range(m):
sigma = gridstd[j]
if np.isnan(sigma):
continue
if flag_uncertainty:
var_model[j] += w_1 * w_2 * sigma**2 * corr
if flag_matrix:
cov_model[j, ii, jj] = sigma ** 2 * corr
cov_model[j, jj, ii] = sigma ** 2 * corr
result = {}
if flag_uncertainty:
std_model = np.sqrt(var_model)
result[&#39;uncertainty&#39;] = std_model
if flag_matrix:
result[&#39;matrix&#39;] = cov_model
return result</code></pre>
</details>
</dd>
<dt id="src.covariance.cov_function_lat"><code class="name flex">
<span>def <span class="ident">cov_function_lat</span></span>(<span>lat1: float, lat2: float, dist: float, theta: float, ny: float, a_0: float, k_a_0_2: float, k_a_0_4: float, k_a_0_6: float, k_a_0_8: float, a_1: float, k_a_1_2: float, k_a_1_4: float, k_a_1_6: float, k_a_1_8: float, c_0: float, k_2: float, k_4: float, k_6: float, k_8: float) ‑> float</span>
</code></dt>
<dd>
<div class="desc"><p>Function to compute covariance function between two points according to
publication Boergens et al. 2020</p>
<h2 id="args">Args</h2>
<dl>
<dt>lat1, lat2: float</dt>
<dt>Latitude of the two points</dt>
<dt><strong><code>d</code></strong></dt>
<dd>float
distance between points</dd>
<dt><strong><code>theta</code></strong></dt>
<dd>float
azimuth angle between points</dd>
<dt><strong><code>ny</code></strong></dt>
<dd>float
order of Bessle function</dd>
<dt><strong><code>a0</code></strong></dt>
<dd>float
anisotropy parameter</dd>
<dt>ka0_2, ka0_4, ka0_6, ka0_8: float</dt>
<dt>Legende polynome Args for a0</dt>
<dt><strong><code>a1</code></strong></dt>
<dd>float
isotropy shape parameter</dd>
<dt>ka1_2, ka1_4, ka1_6, ka1_8: float</dt>
<dt>Legende polynome Args for a1</dt>
<dt><strong><code>c0</code></strong></dt>
<dd>float
amplitude parameter</dd>
</dl>
<p>k2,k4, k6, k8: float
Legende polynome Args for c0</p>