#! /usr/bin/env python
"""Calculate gradients on a raster grid.
Gradient calculators for raster grids
+++++++++++++++++++++++++++++++++++++
.. autosummary::
~landlab.grid.raster_gradients.calc_grad_at_link
~landlab.grid.raster_gradients.calc_grad_across_cell_faces
~landlab.grid.raster_gradients.calc_grad_across_cell_corners
"""
from collections import deque
import numpy as np
from landlab.core.utils import make_optional_arg_into_id_array
from landlab.core.utils import radians_to_degrees
from landlab.utils.decorators import use_field_name_or_array
from .ext.raster_gradient import calc_diff_at_link as _calc_diff_at_link_c
from .ext.raster_gradient import calc_grad_at_link as _calc_grad_at_link_c
[docs]
@use_field_name_or_array("node")
def calc_diff_at_d8(grid, node_values, out=None):
"""Calculate differences of node values over links and diagonals.
Calculates the difference in quantity *node_values* at each link in the
grid.
Parameters
----------
grid : ModelGrid
A ModelGrid.
node_values : ndarray or field name
Values at grid nodes.
out : ndarray, optional
Buffer to hold the result.
Returns
-------
ndarray
Differences across links.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 4), xy_spacing=(4, 3))
>>> z = [
... [60.0, 60.0, 60.0, 60.0],
... [60.0, 60.0, 0.0, 0.0],
... [60.0, 0.0, 0.0, 0.0],
... ]
>>> grid.calc_diff_at_d8(z)
array([ 0., 0., 0., 0., 0., -60., -60., 0., -60., 0., 0.,
-60., 0., 0., -60., 0., 0., 0., 0., -60., 0., -60.,
-60., -60., 0., -60., 0., 0., 0.])
:meta landlab: info-link, gradient
"""
if out is None:
out = np.empty(grid.number_of_d8)
node_values = np.asarray(node_values)
return np.subtract(
node_values[grid.nodes_at_d8[:, 1]],
node_values[grid.nodes_at_d8[:, 0]],
out=out,
)
[docs]
@use_field_name_or_array("node")
def calc_diff_at_diagonal(grid, node_values, out=None):
"""Calculate differences of node values over diagonals.
Calculates the difference in quantity *node_values* at each link in the
grid.
Parameters
----------
grid : ModelGrid
A ModelGrid.
node_values : ndarray or field name
Values at grid nodes.
out : ndarray, optional
Buffer to hold the result.
Returns
-------
ndarray
Differences across links.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 4), xy_spacing=(4, 3))
>>> z = [
... [5.0, 5.0, 5.0, 5.0],
... [5.0, 5.0, 0.0, 0.0],
... [5.0, 0.0, 0.0, 0.0],
... ]
>>> grid.calc_diff_at_diagonal(z)
array([ 0., 0., -5., 0., -5., -5., -5., 0., -5., 0., 0., 0.])
:meta landlab: info-link, gradient
"""
if out is None:
out = np.empty(grid.number_of_diagonals)
node_values = np.asarray(node_values)
return np.subtract(
node_values[grid.nodes_at_diagonal[:, 1]],
node_values[grid.nodes_at_diagonal[:, 0]],
out=out,
)
[docs]
def calc_grad_at_d8(grid, node_values, out=None):
"""Calculate gradients over all diagonals and links.
Parameters
----------
grid : RasterModelGrid
A grid.
node_values : array_like or field name
Values at nodes.
out : ndarray, optional
Buffer to hold result. If `None`, create a new array.
Examples
--------
>>> from landlab import RasterModelGrid
>>> import numpy as np
>>> grid = RasterModelGrid((3, 4), xy_spacing=(4, 3))
>>> z = [
... [60.0, 60.0, 60.0, 60.0],
... [60.0, 60.0, 0.0, 0.0],
... [60.0, 0.0, 0.0, 0.0],
... ]
>>> grid.calc_grad_at_d8(z)
array([ 0., 0., 0., 0., 0., -20., -20., 0., -15., 0., 0.,
-20., 0., 0., -15., 0., 0., 0., 0., -12., 0., -12.,
-12., -12., 0., -12., 0., 0., 0.])
:meta landlab: info-link, gradient
"""
grads = calc_diff_at_d8(grid, node_values, out=out)
grads /= grid.length_of_d8[: grid.number_of_d8]
return grads
[docs]
def calc_grad_at_diagonal(grid, node_values, out=None):
"""Calculate gradients over all diagonals.
Parameters
----------
grid : RasterModelGrid
A grid.
node_values : array_like or field name
Values at nodes.
out : ndarray, optional
Buffer to hold result. If `None`, create a new array.
Examples
--------
>>> from landlab import RasterModelGrid
>>> import numpy as np
>>> grid = RasterModelGrid((3, 4), xy_spacing=(4, 3))
>>> z = [
... [5.0, 5.0, 5.0, 5.0],
... [5.0, 5.0, 0.0, 0.0],
... [5.0, 0.0, 0.0, 0.0],
... ]
>>> grid.calc_grad_at_diagonal(z)
array([ 0., 0., -1., 0., -1., -1., -1., 0., -1., 0., 0., 0.])
:meta landlab: info-link, gradient
"""
grads = calc_diff_at_diagonal(grid, node_values, out=out)
grads /= grid.length_of_diagonal[: grid.number_of_diagonals]
return grads
[docs]
@use_field_name_or_array("node")
def calc_diff_at_link(grid, value_at_node, out=None):
"""Calculate differences in node_values at links.
Parameters
----------
grid : RasterModelGrid
A grid.
value_at_node : array_like or field name
Values at nodes.
out : ndarray, optional
Buffer to hold result. If `None`, create a new array.
Returns
-------
ndarray
Differences of the nodes values for each link.
Examples
--------
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 3))
>>> node_values = [
... [0.0, 0.0, 0.0],
... [1.0, 3.0, 1.0],
... [2.0, 2.0, 2.0],
... ]
>>> grid.calc_diff_at_link(node_values)
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
>>> out = np.empty(grid.number_of_links, dtype=float)
>>> rtn = grid.calc_diff_at_link(node_values, out=out)
>>> rtn is out
True
>>> out
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
>>> grid = RasterModelGrid((3, 3), xy_spacing=(2, 1))
>>> grid.calc_diff_at_link(node_values)
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
>>> _ = grid.add_field("elevation", node_values, at="node")
>>> grid.calc_diff_at_link("elevation")
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
:meta landlab: info-link, gradient
"""
if out is None:
out = grid.empty(at="link")
_calc_diff_at_link_c(grid.shape, np.asarray(value_at_node).reshape(-1), out)
return out
[docs]
@use_field_name_or_array("node")
def calc_grad_at_link(grid, value_at_node, out=None):
"""Calculate gradients in node_values at links.
Parameters
----------
grid : RasterModelGrid
A grid.
value_at_node : array_like or field name
Values at nodes.
out : ndarray, optional
Buffer to hold result. If `None`, create a new array.
Returns
-------
ndarray
Gradients of the nodes values for each link.
Examples
--------
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 3))
>>> node_values = [0.0, 0.0, 0.0, 1.0, 3.0, 1.0, 2.0, 2.0, 2.0]
>>> grid.calc_grad_at_link(node_values)
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
>>> out = np.empty(grid.number_of_links, dtype=float)
>>> rtn = grid.calc_grad_at_link(node_values, out=out)
>>> rtn is out
True
>>> out
array([ 0., 0., 1., 3., 1., 2., -2., 1., -1., 1., 0., 0.])
>>> grid = RasterModelGrid((3, 3), xy_spacing=(2, 1))
>>> grid.calc_grad_at_link(node_values)
array([ 0., 0., 1., 3., 1., 1., -1., 1., -1., 1., 0., 0.])
>>> _ = grid.add_field("elevation", node_values, at="node")
>>> grid.calc_grad_at_link("elevation")
array([ 0., 0., 1., 3., 1., 1., -1., 1., -1., 1., 0., 0.])
:meta landlab: info-link, gradient
"""
if out is None:
out = grid.empty(at="link")
_calc_grad_at_link_c(
grid.shape, (grid.dx, grid.dy), np.asarray(value_at_node).reshape(-1), out
)
return out
[docs]
@use_field_name_or_array("node")
def calc_grad_across_cell_faces(grid, node_values, *args, **kwds):
"""calc_grad_across_cell_faces(grid, node_values, [cell_ids], out=None)
Get gradients across the faces of a cell.
Calculate gradient of the value field provided by *node_values* across
each of the faces of the cells of a grid. The returned gradients are
ordered as right, top, left, and bottom.
Note that the returned gradients are masked to exclude neighbor nodes which
are closed. Beneath the mask is the value -1.
Parameters
----------
grid : RasterModelGrid
Source grid.
node_values : array_like or field name
Quantity to take the gradient of defined at each node.
cell_ids : array_like, optional
If provided, cell ids to measure gradients. Otherwise, find gradients
for all cells.
out : array_like, optional
Alternative output array in which to place the result. Must
be of the same shape and buffer length as the expected output.
Returns
-------
(N, 4) Masked ndarray
Gradients for each face of the cell.
Examples
--------
Create a grid with two cells.
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 4))
>>> x = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 3.0, 3.0, 3.0, 3.0])
A decrease in quantity across a face is a negative gradient.
>>> grid.calc_grad_across_cell_faces(x)
masked_array(data =
[[ 1. 3. 0. 0.]
[ 0. 2. -1. -1.]],
mask =
False,
fill_value = 1e+20)
>>> grid = RasterModelGrid((3, 4), xy_spacing=(1, 2))
>>> grid.calc_grad_across_cell_faces(x)
masked_array(data =
[[ 1. 1.5 0. 0. ]
[ 0. 1. -1. -0.5]],
mask =
False,
fill_value = 1e+20)
:meta landlab: info-face, gradient
"""
padded_node_values = np.empty(node_values.size + 1, dtype=float)
padded_node_values[-1] = grid.BAD_INDEX
padded_node_values[:-1] = node_values
cell_ids = make_optional_arg_into_id_array(grid.number_of_cells, *args)
node_ids = grid.node_at_cell[cell_ids]
neighbors = grid.active_adjacent_nodes_at_node[node_ids]
if grid.BAD_INDEX != -1:
neighbors = np.where(neighbors == grid.BAD_INDEX, -1, neighbors)
values_at_neighbors = padded_node_values[neighbors]
masked_neighbor_values = np.ma.array(
values_at_neighbors, mask=neighbors == grid.BAD_INDEX
)
values_at_nodes = node_values[node_ids].reshape(len(node_ids), 1)
out = np.subtract(masked_neighbor_values, values_at_nodes, **kwds)
out[:, (0, 2)] /= grid.dx
out[:, (1, 3)] /= grid.dy
return out
[docs]
@use_field_name_or_array("node")
def calc_grad_across_cell_corners(grid, node_values, *args, **kwds):
"""calc_grad_across_cell_corners(grid, node_values, [cell_ids], out=None)
Get gradients to diagonally opposite nodes.
Calculate gradient of the value field provided by *node_values* to
the values at diagonally opposite nodes. The returned gradients are
ordered as upper-right, upper-left, lower-left and lower-right.
Parameters
----------
grid : RasterModelGrid
Source grid.
node_values : array_like or field name
Quantity to take the gradient of defined at each node.
cell_ids : array_like, optional
If provided, cell ids to measure gradients. Otherwise, find gradients
for all cells.
out : array_like, optional
Alternative output array in which to place the result. Must
be of the same shape and buffer length as the expected output.
Returns
-------
(N, 4) ndarray
Gradients to each diagonal node.
Examples
--------
Create a grid with two cells.
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 4))
>>> x = np.array([1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 3.0, 3.0, 3.0, 3.0])
A decrease in quantity to a diagonal node is a negative gradient.
>>> from math import sqrt
>>> grid.calc_grad_across_cell_corners(x) * sqrt(2.0)
array([[ 3., 3., 1., 0.],
[ 2., 2., -1., 0.]])
>>> grid = RasterModelGrid((3, 4), xy_spacing=(4, 3))
>>> grid.calc_grad_across_cell_corners(x)
array([[ 0.6, 0.6, 0.2, 0. ],
[ 0.4, 0.4, -0.2, 0. ]])
:meta landlab: info-corner, gradient
"""
cell_ids = make_optional_arg_into_id_array(grid.number_of_cells, *args)
node_ids = grid.node_at_cell[cell_ids]
values_at_diagonals = node_values[grid.diagonal_adjacent_nodes_at_node[node_ids]]
values_at_nodes = node_values[node_ids].reshape(len(node_ids), 1)
out = np.subtract(values_at_diagonals, values_at_nodes, **kwds)
np.divide(out, np.sqrt(grid.dy**2.0 + grid.dx**2.0), out=out)
return out
[docs]
@use_field_name_or_array("node")
def calc_grad_along_node_links(grid, node_values, *args, **kwds):
"""calc_grad_along_node_links(grid, node_values, [cell_ids], out=None)
Get gradients along links touching a node.
Calculate gradient of the value field provided by *node_values* across
each of the faces of the nodes of a grid. The returned gradients are
ordered as right, top, left, and bottom. All returned values follow our
standard sign convention, where a link pointing N or E and increasing in
value is positive, a link pointing S or W and increasing in value is
negative.
Note that the returned gradients are masked to exclude neighbor nodes which
are closed. Beneath the mask is the value -1.
Parameters
----------
grid : RasterModelGrid
Source grid.
node_values : array_like or field name
Quantity to take the gradient of defined at each node.
node_ids : array_like, optional
If provided, node ids to measure gradients. Otherwise, find gradients
for all nodes.
out : array_like, optional
Alternative output array in which to place the result. Must
be of the same shape and buffer length as the expected output.
Returns
-------
(N, 4) Masked ndarray
Gradients for each link of the node. Ordering is E,N,W,S.
Examples
--------
Create a grid with nine nodes.
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 3))
>>> x = np.array([0.0, 0.0, 0.0, 0.0, 1.0, 2.0, 2.0, 2.0, 2.0])
A decrease in quantity across a face is a negative gradient.
>>> grid.calc_grad_along_node_links(x)
masked_array(data =
[[-- -- -- --]
[-- 1.0 -- --]
[-- -- -- --]
[1.0 -- -- --]
[1.0 1.0 1.0 1.0]
[-- -- 1.0 --]
[-- -- -- --]
[-- -- -- 1.0]
[-- -- -- --]],
mask =
[[ True True True True]
[ True False True True]
[ True True True True]
[False True True True]
[False False False False]
[ True True False True]
[ True True True True]
[ True True True False]
[ True True True True]],
fill_value = 1e+20)
>>> grid = RasterModelGrid((3, 3), xy_spacing=(4, 2))
>>> grid.calc_grad_along_node_links(x)
masked_array(data =
[[-- -- -- --]
[-- 0.5 -- --]
[-- -- -- --]
[0.25 -- -- --]
[0.25 0.5 0.25 0.5]
[-- -- 0.25 --]
[-- -- -- --]
[-- -- -- 0.5]
[-- -- -- --]],
mask =
[[ True True True True]
[ True False True True]
[ True True True True]
[False True True True]
[False False False False]
[ True True False True]
[ True True True True]
[ True True True False]
[ True True True True]],
fill_value = 1e+20)
:meta landlab: info-node, info-link, gradient
"""
padded_node_values = np.empty(node_values.size + 1, dtype=float)
padded_node_values[-1] = grid.BAD_INDEX
padded_node_values[:-1] = node_values
node_ids = make_optional_arg_into_id_array(grid.number_of_nodes, *args)
neighbors = grid.active_adjacent_nodes_at_node[node_ids]
values_at_neighbors = padded_node_values[neighbors]
masked_neighbor_values = np.ma.array(
values_at_neighbors, mask=values_at_neighbors == grid.BAD_INDEX
)
values_at_nodes = node_values[node_ids].reshape(len(node_ids), 1)
out = np.ma.empty_like(masked_neighbor_values, dtype=float)
np.subtract(masked_neighbor_values[:, :2], values_at_nodes, out=out[:, :2], **kwds)
np.subtract(values_at_nodes, masked_neighbor_values[:, 2:], out=out[:, 2:], **kwds)
out[:, (0, 2)] /= grid.dx
out[:, (1, 3)] /= grid.dy
return out
[docs]
def calc_unit_normals_at_cell_subtriangles(grid, elevs="topographic__elevation"):
"""Calculate unit normals on a cell.
Calculate the eight unit normal vectors <a, b, c> to the eight
subtriangles of a four-cornered (raster) cell.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
Returns
-------
(n_ENE, n_NNE, n_NNW, n_WNW, n_WSW, n_SSW, n_SSE, n_ESE) :
each a num-cells x length-3 array
Len-8 tuple of the eight unit normal vectors <a, b, c> for the eight
subtriangles in the cell. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((3, 3))
>>> z = mg.node_x**2
>>> eight_tris = mg.calc_unit_normals_at_cell_subtriangles(z)
>>> type(eight_tris) is tuple
True
>>> len(eight_tris)
8
>>> eight_tris[0].shape == (mg.number_of_cells, 3)
True
>>> eight_tris
(array([[-0.9486833 , 0. , 0.31622777]]),
array([[-0.9486833 , 0. , 0.31622777]]),
array([[-0.70710678, 0. , 0.70710678]]),
array([[-0.70710678, 0. , 0.70710678]]),
array([[-0.70710678, 0. , 0.70710678]]),
array([[-0.70710678, 0. , 0.70710678]]),
array([[-0.9486833 , 0. , 0.31622777]]),
array([[-0.9486833 , 0. , 0.31622777]]))
:meta landlab: info-cell, gradient
"""
# identify the grid neigbors at each location
node_at_cell = grid.node_at_cell
# calculate unit normals at all nodes.
(
n_ENE,
n_NNE,
n_NNW,
n_WNW,
n_WSW,
n_SSW,
n_SSE,
n_ESE,
) = _calc_subtriangle_unit_normals_at_node(grid, elevs=elevs)
# return only those at cell.
return (
n_ENE[node_at_cell, :],
n_NNE[node_at_cell, :],
n_NNW[node_at_cell, :],
n_WNW[node_at_cell, :],
n_WSW[node_at_cell, :],
n_SSW[node_at_cell, :],
n_SSE[node_at_cell, :],
n_ESE[node_at_cell, :],
)
def _calc_subtriangle_unit_normals_at_node(grid, elevs="topographic__elevation"):
"""Private Function: Calculate unit normals on subtriangles at all nodes.
Calculate the eight unit normal vectors <a, b, c> to the eight
subtriangles of a four-cornered (raster) cell. Unlike
calc_unit_normals_at_node_subtriangles, this function also
calculated unit normals at the degenerate part-cells around the
boundary.
On the grid boundaries where the cell is not fully defined, the unit normal
is given as <nan, nan, nan>.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
Returns
-------
(n_ENE, n_NNE, n_NNW, n_WNW, n_WSW, n_SSW, n_SSE, n_ESE) :
each a num-nodes x length-3 array
Len-8 tuple of the eight unit normal vectors <a, b, c> for the eight
subtriangles in the cell. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.grid.raster_gradients import _calc_subtriangle_unit_normals_at_node
>>> mg = RasterModelGrid((3, 3))
>>> z = mg.node_x**2
>>> eight_tris = _calc_subtriangle_unit_normals_at_node(mg, z)
>>> type(eight_tris) is tuple
True
>>> len(eight_tris)
8
>>> eight_tris[0].shape == (mg.number_of_nodes, 3)
True
>>> eight_tris[0]
array([[-0.70710678, 0. , 0.70710678],
[-0.9486833 , 0. , 0.31622777],
[ nan, nan, nan],
[-0.70710678, 0. , 0.70710678],
[-0.9486833 , 0. , 0.31622777],
[ nan, nan, nan],
[ nan, nan, nan],
[ nan, nan, nan],
[ nan, nan, nan]])
:meta landlab: info-cell, gradient
"""
try:
z = grid.at_node[elevs]
except TypeError:
z = elevs
# cell has center node I
# orthogonal neighbors P, R, T, V, counter clockwise from East
# diagonal neihbors Q, S, U, W, counter clocwise from North East
# There are 8 subtriangles that can be defined with the following corners
# (starting from the central node, and progressing counter-clockwise).
# ENE: IPQ
# NNE: IQR
# NNW: IRS
# WNW: IST
# WSW: ITU
# SSW: IUV
# SSE: IVW
# ESE: IWP
# There are thus 8 vectors, IP, IQ, IR, IS, IT, IU, IV, IW
# initialized difference matricies for cross product
diff_xyz_IP = np.empty((grid.number_of_nodes, 3)) # East
# ^this is the vector (xP-xI, yP-yI, zP-yI)
diff_xyz_IQ = np.empty((grid.number_of_nodes, 3)) # Northeast
diff_xyz_IR = np.empty((grid.number_of_nodes, 3)) # North
diff_xyz_IS = np.empty((grid.number_of_nodes, 3)) # Northwest
diff_xyz_IT = np.empty((grid.number_of_nodes, 3)) # West
diff_xyz_IU = np.empty((grid.number_of_nodes, 3)) # Southwest
diff_xyz_IV = np.empty((grid.number_of_nodes, 3)) # South
diff_xyz_IW = np.empty((grid.number_of_nodes, 3)) # Southeast
# identify the grid neigbors at each location
node_at_cell = np.arange(grid.number_of_nodes)
P = grid.adjacent_nodes_at_node[node_at_cell, 0]
Q = grid.diagonal_adjacent_nodes_at_node[node_at_cell, 0]
R = grid.adjacent_nodes_at_node[node_at_cell, 1]
S = grid.diagonal_adjacent_nodes_at_node[node_at_cell, 1]
T = grid.adjacent_nodes_at_node[node_at_cell, 2]
U = grid.diagonal_adjacent_nodes_at_node[node_at_cell, 2]
V = grid.adjacent_nodes_at_node[node_at_cell, 3]
W = grid.diagonal_adjacent_nodes_at_node[node_at_cell, 3]
# get x, y, z coordinates for each location
x_I = grid.node_x[node_at_cell]
y_I = grid.node_y[node_at_cell]
z_I = z[node_at_cell]
x_P = grid.node_x[P]
y_P = grid.node_y[P]
z_P = z[P]
x_Q = grid.node_x[Q]
y_Q = grid.node_y[Q]
z_Q = z[Q]
x_R = grid.node_x[R]
y_R = grid.node_y[R]
z_R = z[R]
x_S = grid.node_x[S]
y_S = grid.node_y[S]
z_S = z[S]
x_T = grid.node_x[T]
y_T = grid.node_y[T]
z_T = z[T]
x_U = grid.node_x[U]
y_U = grid.node_y[U]
z_U = z[U]
x_V = grid.node_x[V]
y_V = grid.node_y[V]
z_V = z[V]
x_W = grid.node_x[W]
y_W = grid.node_y[W]
z_W = z[W]
# calculate vectors by differencing
diff_xyz_IP[:, 0] = x_P - x_I
diff_xyz_IP[:, 1] = y_P - y_I
diff_xyz_IP[:, 2] = z_P - z_I
diff_xyz_IQ[:, 0] = x_Q - x_I
diff_xyz_IQ[:, 1] = y_Q - y_I
diff_xyz_IQ[:, 2] = z_Q - z_I
diff_xyz_IR[:, 0] = x_R - x_I
diff_xyz_IR[:, 1] = y_R - y_I
diff_xyz_IR[:, 2] = z_R - z_I
diff_xyz_IS[:, 0] = x_S - x_I
diff_xyz_IS[:, 1] = y_S - y_I
diff_xyz_IS[:, 2] = z_S - z_I
diff_xyz_IT[:, 0] = x_T - x_I
diff_xyz_IT[:, 1] = y_T - y_I
diff_xyz_IT[:, 2] = z_T - z_I
diff_xyz_IU[:, 0] = x_U - x_I
diff_xyz_IU[:, 1] = y_U - y_I
diff_xyz_IU[:, 2] = z_U - z_I
diff_xyz_IV[:, 0] = x_V - x_I
diff_xyz_IV[:, 1] = y_V - y_I
diff_xyz_IV[:, 2] = z_V - z_I
diff_xyz_IW[:, 0] = x_W - x_I
diff_xyz_IW[:, 1] = y_W - y_I
diff_xyz_IW[:, 2] = z_W - z_I
# calculate cross product to get unit normal
# cross product is orthogonal to both vectors, and is the normal
# n = <a, b, c>, where plane is ax + by + cz = d
nhat_ENE = np.cross(diff_xyz_IP, diff_xyz_IQ) # <a, b, c>
nhat_NNE = np.cross(diff_xyz_IQ, diff_xyz_IR)
nhat_NNW = np.cross(diff_xyz_IR, diff_xyz_IS)
nhat_WNW = np.cross(diff_xyz_IS, diff_xyz_IT)
nhat_WSW = np.cross(diff_xyz_IT, diff_xyz_IU)
nhat_SSW = np.cross(diff_xyz_IU, diff_xyz_IV)
nhat_SSE = np.cross(diff_xyz_IV, diff_xyz_IW)
nhat_ESE = np.cross(diff_xyz_IW, diff_xyz_IP)
# now remove the bad subtriangles based on parts of the grid
# make the bad subtriangle of length greater than one.
bad = np.nan
# first, corners:
(northeast, northwest, southwest, southeast) = grid.nodes_at_corners_of_grid
# lower left corner only has NNE and ENE
for array in (nhat_NNW, nhat_WNW, nhat_WSW, nhat_SSW, nhat_SSE, nhat_ESE):
array[southwest] = bad
# lower right corner only has NNW and WNW
for array in (nhat_ENE, nhat_NNE, nhat_WSW, nhat_SSW, nhat_SSE, nhat_ESE):
array[southeast] = bad
# upper left corner only has ESE and SSE
for array in (nhat_ENE, nhat_NNE, nhat_NNW, nhat_WNW, nhat_WSW, nhat_SSW):
array[northwest] = bad
# upper right corner only has WSW and SSW
for array in (nhat_ENE, nhat_NNE, nhat_NNW, nhat_WNW, nhat_SSE, nhat_ESE):
array[northeast] = bad
# next, sizes:
# bottom row only has Norths
bottom = grid.nodes_at_bottom_edge
for array in (nhat_WSW, nhat_SSW, nhat_SSE, nhat_ESE):
array[bottom] = bad
# left side only has Easts
left = grid.nodes_at_left_edge
for array in (nhat_NNW, nhat_WNW, nhat_WSW, nhat_SSW):
array[left] = bad
# top row only has Souths
top = grid.nodes_at_top_edge
for array in (nhat_ENE, nhat_NNE, nhat_NNW, nhat_WNW):
array[top] = bad
# right side only has Wests
right = grid.nodes_at_right_edge
for array in (nhat_ENE, nhat_NNE, nhat_SSE, nhat_ESE):
array[right] = bad
# calculate magnitude of cross product so that the result is a unit normal
nmag_ENE = np.sqrt(np.square(nhat_ENE).sum(axis=1))
nmag_NNE = np.sqrt(np.square(nhat_NNE).sum(axis=1))
nmag_NNW = np.sqrt(np.square(nhat_NNW).sum(axis=1))
nmag_WNW = np.sqrt(np.square(nhat_WNW).sum(axis=1))
nmag_WSW = np.sqrt(np.square(nhat_WSW).sum(axis=1))
nmag_SSW = np.sqrt(np.square(nhat_SSW).sum(axis=1))
nmag_SSE = np.sqrt(np.square(nhat_SSE).sum(axis=1))
nmag_ESE = np.sqrt(np.square(nhat_ESE).sum(axis=1))
# normalize the cross product with its magnitude so it is a unit normal
# instead of a variable length normal.
n_ENE = nhat_ENE / nmag_ENE.reshape(grid.number_of_nodes, 1)
n_NNE = nhat_NNE / nmag_NNE.reshape(grid.number_of_nodes, 1)
n_NNW = nhat_NNW / nmag_NNW.reshape(grid.number_of_nodes, 1)
n_WNW = nhat_WNW / nmag_WNW.reshape(grid.number_of_nodes, 1)
n_WSW = nhat_WSW / nmag_WSW.reshape(grid.number_of_nodes, 1)
n_SSW = nhat_SSW / nmag_SSW.reshape(grid.number_of_nodes, 1)
n_SSE = nhat_SSE / nmag_SSE.reshape(grid.number_of_nodes, 1)
n_ESE = nhat_ESE / nmag_ESE.reshape(grid.number_of_nodes, 1)
return (n_ENE, n_NNE, n_NNW, n_WNW, n_WSW, n_SSW, n_SSE, n_ESE)
[docs]
def calc_slope_at_cell_subtriangles(
grid, elevs="topographic__elevation", subtriangle_unit_normals=None
):
"""Calculate the slope (positive magnitude of gradient) at each of the
eight cell subtriangles.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
subtriangle_unit_normals : tuple of 8 (ncells, 3) arrays (optional)
The unit normal vectors for the eight subtriangles of each cell,
if already known. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
Returns
-------
(s_ENE, s_NNE, s_NNW, s_WNW, s_WSW, s_SSW, s_SSE, s_ESE) :
each a length num-cells array
Len-8 tuple of the slopes (positive gradient magnitude) of each of the
eight cell subtriangles, in radians. Order is from north of east,
counter clockwise to south of east (East North East, North North East,
North North West, West North West, West South West, South South West,
South South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((3, 3))
>>> z = np.array(
... [np.sqrt(3.0), 0.0, 4.0 / 3.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0 / np.sqrt(3.0)]
... )
>>> eight_tris = mg.calc_unit_normals_at_cell_subtriangles(z)
>>> S = mg.calc_slope_at_cell_subtriangles(z, eight_tris)
>>> S0 = mg.calc_slope_at_cell_subtriangles(z)
>>> np.allclose(S, S0)
True
>>> type(S) is tuple
True
>>> len(S)
8
>>> len(S[0]) == mg.number_of_cells
True
>>> np.allclose(S[0], S[1])
True
>>> np.allclose(S[2], S[3])
True
>>> np.allclose(S[4], S[5])
True
>>> np.allclose(S[6], S[7])
True
>>> np.allclose(np.rad2deg(S[0])[0], 30.0)
True
>>> np.allclose(np.rad2deg(S[2])[0], 45.0)
True
>>> np.allclose(np.rad2deg(S[4])[0], 60.0)
True
>>> np.allclose(np.cos(S[6])[0], 3.0 / 5.0)
True
:meta landlab: info-cell, gradient
"""
# calculate all subtriangle slopes
(
s_ENE,
s_NNE,
s_NNW,
s_WNW,
s_WSW,
s_SSW,
s_SSE,
s_ESE,
) = _calc_subtriangle_slopes_at_node(
grid, elevs=elevs, subtriangle_unit_normals=subtriangle_unit_normals
)
# return only those at cell
if s_ENE.shape[0] == grid.number_of_nodes:
node_at_cell = grid.node_at_cell
else:
node_at_cell = np.arange(grid.number_of_cells)
return (
s_ENE[node_at_cell],
s_NNE[node_at_cell],
s_NNW[node_at_cell],
s_WNW[node_at_cell],
s_WSW[node_at_cell],
s_SSW[node_at_cell],
s_SSE[node_at_cell],
s_ESE[node_at_cell],
)
def _calc_subtriangle_slopes_at_node(
grid, elevs="topographic__elevation", subtriangle_unit_normals=None
):
"""Private Function: Calculate subtriangles slope at all nodes.
Calculate the slope (positive magnitude of gradient) at each of the
eight subtriangles, including those at not-full cells along the
boundary.
Those subtriangles that that don't exist because they are on the edge
of the grid have slopes of NAN.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
subtriangle_unit_normals : tuple of 8 (ncells, 3) or (nnodes, 3) arrays
(optional)
The unit normal vectors for the eight subtriangles of each cell or
node,if already known. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
Returns
-------
(s_ENE, s_NNE, s_NNW, s_WNW, s_WSW, s_SSW, s_SSE, s_ESE) :
each a length num-cells array
Len-8 tuple of the slopes (positive gradient magnitude) of each of the
eight cell subtriangles, in radians. Order is from north of east,
counter clockwise to south of east (East North East, North North East,
North North West, West North West, West South West, South South West,
South South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.grid.raster_gradients import (
... _calc_subtriangle_unit_normals_at_node,
... _calc_subtriangle_slopes_at_node,
... )
>>> mg = RasterModelGrid((3, 3))
>>> z = np.array(
... [np.sqrt(3.0), 0.0, 4.0 / 3.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0 / np.sqrt(3.0)]
... )
>>> eight_tris = _calc_subtriangle_unit_normals_at_node(mg, z)
>>> S = _calc_subtriangle_slopes_at_node(mg, z, eight_tris)
>>> S0 = _calc_subtriangle_slopes_at_node(mg, z)
>>> np.allclose(S, S0, equal_nan=True)
True
>>> type(S) is tuple
True
>>> len(S)
8
>>> len(S[0]) == mg.number_of_nodes
True
>>> np.allclose(S[0][mg.core_nodes], S[1][mg.core_nodes])
True
>>> np.allclose(S[2][mg.core_nodes], S[3][mg.core_nodes])
True
>>> np.allclose(S[4][mg.core_nodes], S[5][mg.core_nodes])
True
>>> np.allclose(S[6][mg.core_nodes], S[7][mg.core_nodes])
True
>>> np.allclose(np.rad2deg(S[0][mg.core_nodes]), 30.0)
True
>>> np.allclose(np.rad2deg(S[2][mg.core_nodes]), 45.0)
True
>>> np.allclose(np.rad2deg(S[4])[mg.core_nodes], 60.0)
True
>>> np.allclose(np.cos(S[6])[mg.core_nodes], 3.0 / 5.0)
True
:meta landlab: info-cell, gradient
"""
# verify that subtriangle_unit_normals is of the correct form.
if subtriangle_unit_normals is not None:
assert len(subtriangle_unit_normals) == 8
assert subtriangle_unit_normals[0].shape[1] == 3
assert subtriangle_unit_normals[1].shape[1] == 3
assert subtriangle_unit_normals[2].shape[1] == 3
assert subtriangle_unit_normals[3].shape[1] == 3
assert subtriangle_unit_normals[4].shape[1] == 3
assert subtriangle_unit_normals[5].shape[1] == 3
assert subtriangle_unit_normals[6].shape[1] == 3
assert subtriangle_unit_normals[7].shape[1] == 3
(
n_ENE,
n_NNE,
n_NNW,
n_WNW,
n_WSW,
n_SSW,
n_SSE,
n_ESE,
) = subtriangle_unit_normals
if subtriangle_unit_normals[7].shape[0] == grid.number_of_nodes:
reshape_size = grid.number_of_nodes
elif subtriangle_unit_normals[7].shape[0] == grid.number_of_cells:
reshape_size = grid.number_of_cells
else:
ValueError("Subtriangles must be of lenght nnodes or ncells")
else:
(
n_ENE,
n_NNE,
n_NNW,
n_WNW,
n_WSW,
n_SSW,
n_SSE,
n_ESE,
) = _calc_subtriangle_unit_normals_at_node(grid, elevs)
reshape_size = grid.number_of_nodes
# combine z direction element of all eight so that the arccosine portion
# only takes one function call.
dotprod = np.empty((reshape_size, 8))
dotprod[:, 0] = n_ENE[:, 2] # by definition
dotprod[:, 1] = n_NNE[:, 2]
dotprod[:, 2] = n_NNW[:, 2]
dotprod[:, 3] = n_WNW[:, 2]
dotprod[:, 4] = n_WSW[:, 2]
dotprod[:, 5] = n_SSW[:, 2]
dotprod[:, 6] = n_SSE[:, 2]
dotprod[:, 7] = n_ESE[:, 2]
# take the inverse cosine of the z component to get the slope angle
slopes_at_cell_subtriangles = np.arccos(dotprod) #
# split array into each subtriangle component.
s_ENE = slopes_at_cell_subtriangles[:, 0].reshape(reshape_size)
s_NNE = slopes_at_cell_subtriangles[:, 1].reshape(reshape_size)
s_NNW = slopes_at_cell_subtriangles[:, 2].reshape(reshape_size)
s_WNW = slopes_at_cell_subtriangles[:, 3].reshape(reshape_size)
s_WSW = slopes_at_cell_subtriangles[:, 4].reshape(reshape_size)
s_SSW = slopes_at_cell_subtriangles[:, 5].reshape(reshape_size)
s_SSE = slopes_at_cell_subtriangles[:, 6].reshape(reshape_size)
s_ESE = slopes_at_cell_subtriangles[:, 7].reshape(reshape_size)
return (s_ENE, s_NNE, s_NNW, s_WNW, s_WSW, s_SSW, s_SSE, s_ESE)
[docs]
def calc_aspect_at_cell_subtriangles(
grid, elevs="topographic__elevation", subtriangle_unit_normals=None, unit="degrees"
):
"""Get tuple of arrays of aspect of each of the eight cell subtriangles.
Aspect is returned as radians clockwise of north, unless input parameter
units is set to 'degrees'.
If subtriangle_unit_normals is provided the aspect will be calculated from
these data.
If it is not, it will be derived from elevation data at the nodes,
which can either be a string referring to a grid field (default:
'topographic__elevation'), or an nnodes-long numpy array of the
values themselves.
Parameters
----------
grid : ModelGrid
A ModelGrid.
elevs : str or array (optional)
Node field name or node array of elevations.
If *subtriangle_unit_normals* is not provided, must be set, but unused
otherwise.
subtriangle_unit_normals : tuple of 8 (ncels, 3) arrays (optional)
The unit normal vectors for the eight subtriangles of each cell,
if already known. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
unit : {'degrees', 'radians'}
Controls the unit that the aspect is returned as.
Returns
-------
(a_ENE, a_NNE, a_NNW, a_WNW, a_WSW, a_SSW, a_SSE, a_ESE) :
each a length num-cells array
Len-8 tuple of the aspect of each of the eight cell subtriangles.
Aspect is returned as angle clockwise of north. Units are given as
radians unless input parameter units is set to 'degrees'.
Order is from north of east, counter clockwise to south of east (East
North East, North North East, North North West, West North West, West
South West, South South West, South South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((3, 3))
>>> z = np.array([1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0])
>>> eight_tris = mg.calc_unit_normals_at_cell_subtriangles(z)
>>> A = mg.calc_aspect_at_cell_subtriangles(z, eight_tris)
>>> A0 = mg.calc_aspect_at_cell_subtriangles(z)
>>> np.allclose(A, A0)
True
>>> type(A) is tuple
True
>>> len(A)
8
>>> len(A[0]) == mg.number_of_cells
True
>>> A0
(array([ 180.]), array([ 270.]), array([ 90.]), array([ 180.]),
array([ 0.]), array([ 90.]), array([ 270.]), array([ 0.]))
:meta landlab: info-cell, surface
"""
# calculate all subtriangle slopes
(
angle_ENE,
angle_NNE,
angle_NNW,
angle_WNW,
angle_WSW,
angle_SSW,
angle_SSE,
angle_ESE,
) = _calc_subtriangle_aspect_at_node(
grid, elevs=elevs, subtriangle_unit_normals=subtriangle_unit_normals, unit=unit
)
# return only those at cell
if angle_ESE.shape[0] == grid.number_of_nodes:
node_at_cell = grid.node_at_cell
else:
node_at_cell = np.arange(grid.number_of_cells)
if unit == "degrees" or unit == "radians":
return (
angle_ENE[node_at_cell],
angle_NNE[node_at_cell],
angle_NNW[node_at_cell],
angle_WNW[node_at_cell],
angle_WSW[node_at_cell],
angle_SSW[node_at_cell],
angle_SSE[node_at_cell],
angle_ESE[node_at_cell],
)
else:
raise TypeError("unit must be 'degrees' or 'radians'")
def _calc_subtriangle_aspect_at_node(
grid, elevs="topographic__elevation", subtriangle_unit_normals=None, unit="degrees"
):
"""Private Function: Aspect of subtriangles at node.
This function calculates the aspect of all subtriangles, including those
that are at noded without cells (on the boundaries).
Aspect is returned as radians clockwise of north, unless input parameter
units is set to 'degrees'.
If subtriangle_unit_normals is provided the aspect will be calculated from
these data.
If it is not, it will be derived from elevation data at the nodes,
which can either be a string referring to a grid field (default:
'topographic__elevation'), or an nnodes-long numpy array of the
values themselves.
Parameters
----------
grid : ModelGrid
A ModelGrid.
elevs : str or array (optional)
Node field name or node array of elevations.
If *subtriangle_unit_normals* is not provided, must be set, but unused
otherwise.
subtriangle_unit_normals : tuple of 8 (ncells, 3) or (nnodes, 3) arrays
(optional)
The unit normal vectors for the eight subtriangles of each cell or
node,if already known. Order is from north of east, counter
clockwise to south of east (East North East, North North East, North
North West, West North West, West South West, South South West, South
South East, East South East).
unit : {'degrees', 'radians'}
Controls the unit that the aspect is returned as.
Returns
-------
(a_ENE, a_NNE, a_NNW, a_WNW, a_WSW, a_SSW, a_SSE, a_ESE) :
each a length num-cells array
Len-8 tuple of the aspect of each of the eight cell subtriangles.
Aspect is returned as angle clockwise of north. Units are given as
radians unless input parameter units is set to 'degrees'.
Order is from north of east, counter clockwise to south of east (East
North East, North North East, North North West, West North West, West
South West, South South West, South South East, East South East).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> from landlab.grid.raster_gradients import (
... _calc_subtriangle_unit_normals_at_node,
... _calc_subtriangle_aspect_at_node,
... )
>>> mg = RasterModelGrid((3, 3))
>>> z = np.array([1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0])
>>> eight_tris = _calc_subtriangle_unit_normals_at_node(mg, z)
>>> A = _calc_subtriangle_aspect_at_node(mg, z, eight_tris)
>>> A0 = _calc_subtriangle_aspect_at_node(mg, z)
>>> np.allclose(A, A0, equal_nan=True)
True
>>> type(A) is tuple
True
>>> len(A)
8
>>> len(A[0]) == mg.number_of_nodes
True
>>> A0
(array([ 90., 315., nan, 90., 180., nan, nan, nan, nan]),
array([ 0., 90., nan, 135., 270., nan, nan, nan, nan]),
array([ nan, 90., 0., nan, 90., 225., nan, nan, nan]),
array([ nan, 45., 270., nan, 180., 90., nan, nan, nan]),
array([ nan, nan, nan, nan, 0., 90., nan, 135., 270.]),
array([ nan, nan, nan, nan, 90., 315., nan, 90., 180.]),
array([ nan, nan, nan, 45., 270., nan, 180., 90., nan]),
array([ nan, nan, nan, 270., 0., nan, 90., 225., nan]))
:meta landlab: info-cell, surface
"""
# verify that subtriangle_unit_normals is of the correct form.
if subtriangle_unit_normals is not None:
assert len(subtriangle_unit_normals) == 8
assert subtriangle_unit_normals[0].shape[1] == 3
assert subtriangle_unit_normals[1].shape[1] == 3
assert subtriangle_unit_normals[2].shape[1] == 3
assert subtriangle_unit_normals[3].shape[1] == 3
assert subtriangle_unit_normals[4].shape[1] == 3
assert subtriangle_unit_normals[5].shape[1] == 3
assert subtriangle_unit_normals[6].shape[1] == 3
assert subtriangle_unit_normals[7].shape[1] == 3
if subtriangle_unit_normals[7].shape[0] == grid.number_of_nodes:
reshape_size = grid.number_of_nodes
elif subtriangle_unit_normals[7].shape[0] == grid.number_of_cells:
reshape_size = grid.number_of_cells
else:
ValueError("Subtriangles must be of lenght nnodes or ncells")
(
n_ENE,
n_NNE,
n_NNW,
n_WNW,
n_WSW,
n_SSW,
n_SSE,
n_ESE,
) = subtriangle_unit_normals
# otherwise create it.
else:
(
n_ENE,
n_NNE,
n_NNW,
n_WNW,
n_WSW,
n_SSW,
n_SSE,
n_ESE,
) = _calc_subtriangle_unit_normals_at_node(grid, elevs)
reshape_size = grid.number_of_nodes
# calculate the aspect as an angle ccw from the x axis (math angle)
angle_from_x_ccw_ENE = np.reshape(
np.arctan2(n_ENE[:, 1], n_ENE[:, 0]), reshape_size
)
angle_from_x_ccw_NNE = np.reshape(
np.arctan2(n_NNE[:, 1], n_NNE[:, 0]), reshape_size
)
angle_from_x_ccw_NNW = np.reshape(
np.arctan2(n_NNW[:, 1], n_NNW[:, 0]), reshape_size
)
angle_from_x_ccw_WNW = np.reshape(
np.arctan2(n_WNW[:, 1], n_WNW[:, 0]), reshape_size
)
angle_from_x_ccw_WSW = np.reshape(
np.arctan2(n_WSW[:, 1], n_WSW[:, 0]), reshape_size
)
angle_from_x_ccw_SSW = np.reshape(
np.arctan2(n_SSW[:, 1], n_SSW[:, 0]), reshape_size
)
angle_from_x_ccw_SSE = np.reshape(
np.arctan2(n_SSE[:, 1], n_SSE[:, 0]), reshape_size
)
angle_from_x_ccw_ESE = np.reshape(
np.arctan2(n_ESE[:, 1], n_ESE[:, 0]), reshape_size
)
# convert reference from math angle to angles clockwise from north
# return as either radians or degrees depending on unit.
if unit == "degrees":
return (
radians_to_degrees(angle_from_x_ccw_ENE),
radians_to_degrees(angle_from_x_ccw_NNE),
radians_to_degrees(angle_from_x_ccw_NNW),
radians_to_degrees(angle_from_x_ccw_WNW),
radians_to_degrees(angle_from_x_ccw_WSW),
radians_to_degrees(angle_from_x_ccw_SSW),
radians_to_degrees(angle_from_x_ccw_SSE),
radians_to_degrees(angle_from_x_ccw_ESE),
)
elif unit == "radians":
angle_from_north_cw_ENE = (5.0 * np.pi / 2.0 - angle_from_x_ccw_ENE) % (
2.0 * np.pi
)
angle_from_north_cw_NNE = (5.0 * np.pi / 2.0 - angle_from_x_ccw_NNE) % (
2.0 * np.pi
)
angle_from_north_cw_NNW = (5.0 * np.pi / 2.0 - angle_from_x_ccw_NNW) % (
2.0 * np.pi
)
angle_from_north_cw_WNW = (5.0 * np.pi / 2.0 - angle_from_x_ccw_WNW) % (
2.0 * np.pi
)
angle_from_north_cw_WSW = (5.0 * np.pi / 2.0 - angle_from_x_ccw_WSW) % (
2.0 * np.pi
)
angle_from_north_cw_SSW = (5.0 * np.pi / 2.0 - angle_from_x_ccw_SSW) % (
2.0 * np.pi
)
angle_from_north_cw_SSE = (5.0 * np.pi / 2.0 - angle_from_x_ccw_SSE) % (
2.0 * np.pi
)
angle_from_north_cw_ESE = (5.0 * np.pi / 2.0 - angle_from_x_ccw_ESE) % (
2.0 * np.pi
)
return (
angle_from_north_cw_ENE,
angle_from_north_cw_NNE,
angle_from_north_cw_NNW,
angle_from_north_cw_WNW,
angle_from_north_cw_WSW,
angle_from_north_cw_SSW,
angle_from_north_cw_SSE,
angle_from_north_cw_ESE,
)
else:
raise TypeError("unit must be 'degrees' or 'radians'")
[docs]
def calc_unit_normals_at_patch_subtriangles(grid, elevs="topographic__elevation"):
"""Calculate unit normals on a patch.
Calculate the four unit normal vectors <a, b, c> to the four possible
subtriangles of a four-cornered (raster) patch.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
Returns
-------
(n_TR, n_TL, n_BL, n_BR) : each a num-patches x length-3 array
Len-4 tuple of the four unit normal vectors <a, b, c> for the four
possible subtriangles in the patch. Order is (topright, topleft,
bottomleft, bottomright).
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((4, 5))
>>> z = mg.node_x**2
>>> four_tris = mg.calc_unit_normals_at_patch_subtriangles(z)
>>> type(four_tris) is tuple
True
>>> len(four_tris)
4
>>> np.allclose(four_tris[0], four_tris[1])
True
>>> np.allclose(four_tris[2], four_tris[3])
True
>>> np.allclose(four_tris[0], four_tris[2])
True
>>> np.allclose(np.square(four_tris[0]).sum(axis=1), 1.0)
True
>>> four_tris[0]
array([[-0.70710678, 0. , 0.70710678],
[-0.9486833 , 0. , 0.31622777],
[-0.98058068, 0. , 0.19611614],
[-0.98994949, 0. , 0.14142136],
[-0.70710678, 0. , 0.70710678],
[-0.9486833 , 0. , 0.31622777],
[-0.98058068, 0. , 0.19611614],
[-0.98994949, 0. , 0.14142136],
[-0.70710678, 0. , 0.70710678],
[-0.9486833 , 0. , 0.31622777],
[-0.98058068, 0. , 0.19611614],
[-0.98994949, 0. , 0.14142136]])
:meta landlab: info-patch, gradient
"""
try:
z = grid.at_node[elevs]
except TypeError:
z = elevs
# conceptualize patches as TWO sets of 3 nodes
# the corners are PQRS, CC from NE
diff_xyz_PQ = np.empty((grid.number_of_patches, 3)) # TOP
# ^this is the vector (xQ-xP, yQ-yP, zQ-yP)
diff_xyz_PS = np.empty((grid.number_of_patches, 3)) # RIGHT
# we have RS and QR implicitly in PQ and PS - but store them too
diff_xyz_RS = np.empty((grid.number_of_patches, 3)) # BOTTOM
diff_xyz_QR = np.empty((grid.number_of_patches, 3)) # LEFT
P = grid.nodes_at_patch[:, 0]
Q = grid.nodes_at_patch[:, 1]
R = grid.nodes_at_patch[:, 2]
S = grid.nodes_at_patch[:, 3]
x_P = grid.node_x[P]
y_P = grid.node_y[P]
z_P = z[P]
x_Q = grid.node_x[Q]
y_Q = grid.node_y[Q]
z_Q = z[Q]
x_R = grid.node_x[R]
y_R = grid.node_y[R]
z_R = z[R]
x_S = grid.node_x[S]
y_S = grid.node_y[S]
z_S = z[S]
diff_xyz_PQ[:, 0] = x_Q - x_P
diff_xyz_PQ[:, 1] = y_Q - y_P
diff_xyz_PQ[:, 2] = z_Q - z_P
diff_xyz_PS[:, 0] = x_S - x_P
diff_xyz_PS[:, 1] = y_S - y_P
diff_xyz_PS[:, 2] = z_S - z_P
diff_xyz_RS[:, 0] = x_S - x_R
diff_xyz_RS[:, 1] = y_S - y_R
diff_xyz_RS[:, 2] = z_S - z_R
diff_xyz_QR[:, 0] = x_R - x_Q
diff_xyz_QR[:, 1] = y_R - y_Q
diff_xyz_QR[:, 2] = z_R - z_Q
# make the other ones
# cross product is orthogonal to both vectors, and is the normal
# n = <a, b, c>, where plane is ax + by + cz = d
nhat_topleft = np.cross(diff_xyz_PQ, diff_xyz_QR) # <a, b, c>
nhat_bottomright = np.cross(diff_xyz_PS, diff_xyz_RS)
nhat_topright = np.cross(diff_xyz_PQ, diff_xyz_PS)
nhat_bottomleft = np.cross(diff_xyz_QR, diff_xyz_RS)
nmag_topleft = np.sqrt(np.square(nhat_topleft).sum(axis=1))
nmag_bottomright = np.sqrt(np.square(nhat_bottomright).sum(axis=1))
nmag_topright = np.sqrt(np.square(nhat_topright).sum(axis=1))
nmag_bottomleft = np.sqrt(np.square(nhat_bottomleft).sum(axis=1))
n_TR = nhat_topright / nmag_topright.reshape(grid.number_of_patches, 1)
n_TL = nhat_topleft / nmag_topleft.reshape(grid.number_of_patches, 1)
n_BL = nhat_bottomleft / nmag_bottomleft.reshape(grid.number_of_patches, 1)
n_BR = nhat_bottomright / nmag_bottomright.reshape(grid.number_of_patches, 1)
return (n_TR, n_TL, n_BL, n_BR)
[docs]
def calc_slope_at_patch(
grid,
elevs="topographic__elevation",
ignore_closed_nodes=True,
subtriangle_unit_normals=None,
):
"""Calculate the slope (positive magnitude of gradient) at raster patches.
Returns the mean of the slopes of the four possible patch subtriangles.
If ignore_closed_nodes is True, closed nodes do not affect slope
calculations. If more than one closed node is present in a patch, the
patch slope is set to zero.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
ignore_closed_nodes : bool
If True, do not incorporate values at closed nodes into the calc.
subtriangle_unit_normals : tuple of 4 (npatches, 3) arrays (optional)
The unit normal vectors for the four subtriangles of each patch,
if already known. Order is TR, TL, BL, BR.
Returns
-------
slopes_at_patch : n_patches-long array
The slope (positive gradient magnitude) of each patch, in radians.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((4, 5))
>>> z = mg.node_x
>>> S = mg.calc_slope_at_patch(elevs=z)
>>> S.size == mg.number_of_patches
True
>>> np.allclose(S, np.pi / 4.0)
True
>>> z = mg.node_y**2
>>> mg.calc_slope_at_patch(elevs=z).reshape((3, 4))
array([[ 0.78539816, 0.78539816, 0.78539816, 0.78539816],
[ 1.24904577, 1.24904577, 1.24904577, 1.24904577],
[ 1.37340077, 1.37340077, 1.37340077, 1.37340077]])
>>> z = mg.node_x.copy()
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, True)
>>> mg.status_at_node[11] = mg.BC_NODE_IS_CLOSED
>>> mg.status_at_node[9] = mg.BC_NODE_IS_FIXED_VALUE
>>> z[11] = 100.0 # this should get ignored now
>>> z[9] = 2.0 # this should be felt by patch 7 only
>>> mg.calc_slope_at_patch(elevs=z, ignore_closed_nodes=True).reshape(
... (3, 4)
... ) * 4.0 / np.pi
array([[ 0., 0., 0., 0.],
[ 0., 1., 1., 1.],
[ 0., 0., 0., 0.]])
:meta landlab: info-patch, gradient
"""
if subtriangle_unit_normals is not None:
assert len(subtriangle_unit_normals) == 4
assert subtriangle_unit_normals[0].shape[1] == 3
assert subtriangle_unit_normals[1].shape[1] == 3
assert subtriangle_unit_normals[2].shape[1] == 3
assert subtriangle_unit_normals[3].shape[1] == 3
n_TR, n_TL, n_BL, n_BR = subtriangle_unit_normals
else:
n_TR, n_TL, n_BL, n_BR = grid.calc_unit_normals_at_patch_subtriangles(elevs)
dotprod_TL = n_TL[:, 2] # by definition
dotprod_BR = n_BR[:, 2]
dotprod_TR = n_TR[:, 2]
dotprod_BL = n_BL[:, 2]
slopes_at_patch_TL = np.arccos(dotprod_TL) # 1 node order
slopes_at_patch_BR = np.arccos(dotprod_BR) # 3
slopes_at_patch_TR = np.arccos(dotprod_TR) # 0
slopes_at_patch_BL = np.arccos(dotprod_BL) # 2
if ignore_closed_nodes:
badnodes = grid.status_at_node[grid.nodes_at_patch] == grid.BC_NODE_IS_CLOSED
tot_bad = badnodes.sum(axis=1)
tot_tris = 4.0 - 3.0 * (tot_bad > 0) # 4 where all good, 1 where not
# now shut down the bad tris. Remember, one bad node => 3 bad tris.
# anywhere where badnodes > 1 will have zero from summing, so div by 1
# assert np.all(np.logical_or(np.isclose(tot_tris, 4.),
# np.isclose(tot_tris, 1.)))
corners_rot = deque(
[
slopes_at_patch_BR,
slopes_at_patch_TR,
slopes_at_patch_TL,
slopes_at_patch_BL,
]
)
# note initial offset so we are centered around TR on first slice
for i in range(4):
for j in range(3):
(corners_rot[j])[badnodes[:, i]] = 0.0
corners_rot.rotate(-1)
else:
tot_tris = 4.0
mean_slope_at_patch = (
slopes_at_patch_TR
+ slopes_at_patch_TL
+ slopes_at_patch_BL
+ slopes_at_patch_BR
) / tot_tris
return mean_slope_at_patch
[docs]
def calc_grad_at_patch(
grid,
elevs="topographic__elevation",
ignore_closed_nodes=True,
subtriangle_unit_normals=None,
slope_magnitude=None,
):
"""Calculate the components of the gradient of each raster patch.
Returns the mean gradient of the four possible patch subtriangles,
in radians.
If ignore_closed_nodes is True, closed nodes do not affect gradient
calculations. If more than one closed node is present in a patch, the
patch gradients in both x and y directions are set to zero.
Parameters
----------
grid : RasterModelGrid
A grid.
elevs : str or ndarray, optional
Field name or array of node values.
ignore_closed_nodes : bool
If True, do not incorporate values at closed nodes into the calc.
subtriangle_unit_normals : tuple of 4 (npatches, 3) arrays (optional)
The unit normal vectors for the four subtriangles of each patch,
if already known. Order is TR, TL, BL, BR.
slope_magnitude : array with size num_patches (optional)
The mean slope of each patch, if already known. Units must be the
same as provided here!
Returns
-------
gradient_tuple : (x_component_at_patch, y_component_at_patch)
Len-2 tuple of arrays giving components of gradient in the x and y
directions, in the units of *radians*.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((4, 5))
>>> z = mg.node_y
>>> (x_grad, y_grad) = mg.calc_grad_at_patch(elevs=z)
>>> np.allclose(y_grad, np.pi / 4.0)
True
>>> np.allclose(x_grad, 0.0)
True
>>> z = mg.node_x.copy()
>>> mg.set_closed_boundaries_at_grid_edges(True, True, True, True)
>>> mg.status_at_node[11] = mg.BC_NODE_IS_CLOSED
>>> mg.status_at_node[[9, 2]] = mg.BC_NODE_IS_FIXED_VALUE
>>> z[11] = 100.0 # this should get ignored now
>>> z[9] = 2.0 # this should be felt by patch 7 only
>>> z[2] = 1.0 # should be felt by patches 1 and 2
>>> xgrad, ygrad = mg.calc_grad_at_patch(elevs=z, ignore_closed_nodes=True)
>>> (xgrad.reshape((3, 4)) * 4.0 / np.pi)[1, 1:]
array([ 1., 1., -1.])
>>> np.allclose(ygrad[1:3], xgrad[1:3])
True
:meta landlab: info-patch, gradient
"""
if subtriangle_unit_normals is not None:
assert len(subtriangle_unit_normals) == 4
assert subtriangle_unit_normals[0].shape[1] == 3
assert subtriangle_unit_normals[1].shape[1] == 3
assert subtriangle_unit_normals[2].shape[1] == 3
assert subtriangle_unit_normals[3].shape[1] == 3
n_TR, n_TL, n_BL, n_BR = subtriangle_unit_normals
else:
n_TR, n_TL, n_BL, n_BR = grid.calc_unit_normals_at_patch_subtriangles(elevs)
if slope_magnitude is not None:
assert slope_magnitude.size == grid.number_of_patches
slopes_at_patch = slope_magnitude
else:
slopes_at_patch = grid.calc_slope_at_patch(
elevs=elevs,
ignore_closed_nodes=ignore_closed_nodes,
subtriangle_unit_normals=(n_TR, n_TL, n_BL, n_BR),
)
if ignore_closed_nodes:
badnodes = grid.status_at_node[grid.nodes_at_patch] == grid.BC_NODE_IS_CLOSED
corners_rot = deque([n_BR, n_TR, n_TL, n_BL])
# note initial offset so we are centered around TR on first slice
for i in range(4):
for j in range(3):
(corners_rot[j])[badnodes[:, i], :] = 0.0
corners_rot.rotate(-1)
n_sum_x = n_TR[:, 0] + n_TL[:, 0] + n_BL[:, 0] + n_BR[:, 0]
n_sum_y = n_TR[:, 1] + n_TL[:, 1] + n_BL[:, 1] + n_BR[:, 1]
theta_sum = np.arctan2(-n_sum_y, -n_sum_x)
x_slope_patches = np.cos(theta_sum) * slopes_at_patch
y_slope_patches = np.sin(theta_sum) * slopes_at_patch
return (x_slope_patches, y_slope_patches)
[docs]
def calc_slope_at_node(
grid,
elevs="topographic__elevation",
method="patch_mean",
ignore_closed_nodes=True,
return_components=False,
):
"""Array of slopes at nodes, averaged over neighboring patches.
Produces a value for node slope (i.e., mean gradient magnitude)
at each node in a manner analogous to a GIS-style slope map.
If method=='patch_mean', it averages the gradient on each of the
patches surrounding the node; if method=='Horn', it returns the
resolved slope direction. Directional information can still be
returned through use of the return_components keyword.
All values are returned in radians, including the components;
take the tan to recover the rise/run.
Note that under these definitions, it is not always true that::
mag, cmp = mg.calc_slope_at_node(z)
mag**2 == cmp[0]**2 + cmp[1]**2 # only if method=='Horn'
If ignore_closed_nodes is False, all proximal elevation values will be used
in the calculation. If True, only unclosed nodes are used.
This is a verion of this code specialized for a raster. It subdivides
the four square patches around each node into subtriangles,
in order to ensure more correct solutions that incorporate equally
weighted information from all surrounding nodes on rough surfaces.
Parameters
----------
elevs : str or ndarray, optional
Field name or array of node values.
method : {'patch_mean', 'Horn'}
Controls the slope algorithm. Current options are 'patch_mean',
which takes the mean slope of each pf the four neighboring
square patches, and 'Horn', which is the standard ArcGIS slope
algorithm. These produce very similar solutions; the Horn method
gives a vector mean and the patch_mean gives a scalar mean.
ignore_closed_nodes : bool
If True, do not incorporate values at closed nodes into the calc.
return_components : bool
If True, return a tuple, (array_of_magnitude,
(array_of_slope_x_radians, array_of_slope_y_radians)).
If false, return an array of floats of the slope magnitude.
Returns
-------
float array or length-2 tuple of float arrays
If return_components, returns (array_of_magnitude,
(array_of_slope_x_radians, array_of_slope_y_radians)).
If not return_components, returns an array of slope magnitudes.
Examples
--------
>>> import numpy as np
>>> from landlab import RadialModelGrid, RasterModelGrid
>>> mg = RasterModelGrid((5, 5))
>>> z = mg.node_x
>>> slopes = mg.calc_slope_at_node(elevs=z)
>>> np.allclose(slopes, np.pi / 4.0)
True
>>> mg = RasterModelGrid((4, 5), xy_spacing=2.0)
>>> z = -mg.node_y
>>> slope_mag, cmp = mg.calc_slope_at_node(elevs=z, return_components=True)
>>> np.allclose(slope_mag, np.pi / 4.0)
True
>>> np.allclose(cmp[0], 0.0)
True
>>> np.allclose(cmp[1], -np.pi / 4.0)
True
>>> mg = RasterModelGrid((4, 4))
>>> z = mg.node_x**2 + mg.node_y**2
>>> slopes, cmp = mg.calc_slope_at_node(z, return_components=True)
>>> slopes
array([ 0.95531662, 1.10991779, 1.32082849, 1.37713803, 1.10991779,
1.20591837, 1.3454815 , 1.38904403, 1.32082849, 1.3454815 ,
1.39288142, 1.41562833, 1.37713803, 1.38904403, 1.41562833,
1.43030663])
Check radial symmetry.
>>> np.allclose(cmp[0].reshape((4, 4))[:, 0], cmp[1].reshape((4, 4))[0, :])
True
:meta landlab: info-node, gradient, surface
"""
if method not in ("patch_mean", "Horn"):
raise ValueError("method name not understood")
try:
patches_at_node = grid.patches_at_node()
except TypeError: # was a property, not a fn (=> new style)
if not ignore_closed_nodes:
patches_at_node = np.ma.masked_where(
grid.patches_at_node == -1, grid.patches_at_node, copy=False
)
else:
patches_at_node = np.ma.masked_where(
np.logical_not(grid.patches_present_at_node),
grid.patches_at_node,
copy=False,
)
# now, we also want to mask any "closed" patches (any node closed)
closed_patches = (
grid.status_at_node[grid.nodes_at_patch] == grid.BC_NODE_IS_CLOSED
).sum(axis=1) > 0
closed_patch_mask = np.logical_or(
patches_at_node.mask, closed_patches[patches_at_node.data]
)
if method == "patch_mean":
n_TR, n_TL, n_BL, n_BR = grid.calc_unit_normals_at_patch_subtriangles(elevs)
mean_slope_at_patches = grid.calc_slope_at_patch(
elevs=elevs,
ignore_closed_nodes=ignore_closed_nodes,
subtriangle_unit_normals=(n_TR, n_TL, n_BL, n_BR),
)
# now CAREFUL - patches_at_node is MASKED
slopes_at_node_unmasked = mean_slope_at_patches[patches_at_node]
slopes_at_node_masked = np.ma.array(
slopes_at_node_unmasked, mask=closed_patch_mask
)
slope_mag = np.mean(slopes_at_node_masked, axis=1).data
if return_components:
(x_slope_patches, y_slope_patches) = grid.calc_grad_at_patch(
elevs=elevs,
ignore_closed_nodes=ignore_closed_nodes,
subtriangle_unit_normals=(n_TR, n_TL, n_BL, n_BR),
slope_magnitude=mean_slope_at_patches,
)
x_slope_unmasked = x_slope_patches[patches_at_node]
x_slope_masked = np.ma.array(x_slope_unmasked, mask=closed_patch_mask)
x_slope = np.mean(x_slope_masked, axis=1).data
y_slope_unmasked = y_slope_patches[patches_at_node]
y_slope_masked = np.ma.array(y_slope_unmasked, mask=closed_patch_mask)
y_slope = np.mean(y_slope_masked, axis=1).data
mean_grad_x = x_slope
mean_grad_y = y_slope
elif method == "Horn":
z = np.empty(grid.number_of_nodes + 1, dtype=float)
mean_grad_x = grid.empty(at="node", dtype=float)
mean_grad_y = grid.empty(at="node", dtype=float)
z[-1] = 0.0
try:
z[:-1] = grid.at_node[elevs]
except TypeError:
z[:-1] = elevs
# proof code for bad indexing:
diags = grid.diagonal_adjacent_nodes_at_node.copy() # LL order
orthos = grid.adjacent_nodes_at_node.copy()
# these have closed node neighbors...
for dirs in (diags, orthos):
dirs[dirs == grid.BAD_INDEX] = -1 # indexing to work
# now make an array like patches_at_node to store the interim calcs
patch_slopes_x = np.ma.zeros(patches_at_node.shape, dtype=float)
patch_slopes_y = np.ma.zeros(patches_at_node.shape, dtype=float)
diff_E = z[orthos[:, 0]] - z[:-1]
diff_W = z[:-1] - z[orthos[:, 2]]
diff_N = z[orthos[:, 1]] - z[:-1]
diff_S = z[:-1] - z[orthos[:, 3]]
patch_slopes_x[:, 0] = z[diags[:, 0]] - z[orthos[:, 1]] + diff_E
patch_slopes_x[:, 1] = z[orthos[:, 1]] - z[diags[:, 1]] + diff_W
patch_slopes_x[:, 2] = z[orthos[:, 3]] - z[diags[:, 2]] + diff_W
patch_slopes_x[:, 3] = z[diags[:, 3]] - z[orthos[:, 3]] + diff_E
patch_slopes_y[:, 0] = z[diags[:, 0]] - z[orthos[:, 0]] + diff_N
patch_slopes_y[:, 1] = z[diags[:, 1]] - z[orthos[:, 2]] + diff_N
patch_slopes_y[:, 2] = z[orthos[:, 2]] - z[diags[:, 2]] + diff_S
patch_slopes_y[:, 3] = z[orthos[:, 0]] - z[diags[:, 3]] + diff_S
patch_slopes_x /= 2.0 * grid.dx
patch_slopes_y /= 2.0 * grid.dy
patch_slopes_x.mask = closed_patch_mask
patch_slopes_y.mask = closed_patch_mask
mean_grad_x = patch_slopes_x.mean(axis=1).data
mean_grad_y = patch_slopes_y.mean(axis=1).data
slope_mag = np.arctan(np.sqrt(np.square(mean_grad_x) + np.square(mean_grad_y)))
if return_components:
mean_grad_x = np.arctan(mean_grad_x)
mean_grad_y = np.arctan(mean_grad_y)
if return_components:
return slope_mag, (mean_grad_x, mean_grad_y)
else:
return slope_mag