#! /usr/bin/env python
"""Calculate gradients of quantities over links.
Gradient calculation functions
+++++++++++++++++++++++
.. autosummary::
:toctree: generated/
~landlab.grid.gradients.calc_grad_at_active_link
~landlab.grid.gradients.calc_grad_at_link
~landlab.grid.gradients.calculate_gradients_at_faces
~landlab.grid.gradients.calculate_diff_at_links
~landlab.grid.gradients.calculate_diff_at_active_links
"""
import numpy as np
from landlab.utils.decorators import use_field_name_or_array, deprecated
from landlab.core.utils import radians_to_degrees
from landlab.grid.base import CLOSED_BOUNDARY
@use_field_name_or_array('node')
def calc_grad_at_link(grid, node_values, out=None):
"""Calculate gradients of node values at links.
Calculates the gradient in `node_values` at each link in the grid,
returning an array of length `number_of_links`.
Construction::
calc_grad_at_link(grid, node_values, out=None)
Parameters
----------
grid : ModelGrid
A ModelGrid.
node_values : ndarray or field name (x number of nodes)
Values at grid nodes.
out : ndarray, optional (x number of links)
Buffer to hold the result.
Returns
-------
ndarray
Gradients across active links.
Examples
--------
>>> from landlab import RasterModelGrid
>>> rg = RasterModelGrid(3, 4, 10.0)
>>> z = rg.add_zeros('node', 'topographic__elevation')
>>> z[5] = 50.0
>>> z[6] = 36.0
>>> calc_grad_at_link(rg, z) # there are 17 links
array([ 0. , 0. , 0. , 0. , 5. , 3.6, 0. , 5. , -1.4, -3.6, 0. ,
-5. , -3.6, 0. , 0. , 0. , 0. ])
>>> from landlab import HexModelGrid
>>> hg = HexModelGrid(3, 3, 10.0)
>>> z = hg.add_zeros('node', 'topographic__elevation', noclobber=False)
>>> z[4] = 50.0
>>> z[5] = 36.0
>>> calc_grad_at_link(hg, z) # there are 11 faces
array([ 0. , 0. , 0. , 5. , 5. , 3.6, 3.6, 0. , 5. , -1.4, -3.6,
0. , -5. , -5. , -3.6, -3.6, 0. , 0. , 0. ])
LLCATS: LINF GRAD
"""
if out is None:
out = grid.empty(at='link')
return np.divide(node_values[grid.node_at_link_head] -
node_values[grid.node_at_link_tail],
grid.length_of_link, out=out)
@deprecated(use='calc_grad_at_link', version='1.0beta')
def calc_grad_of_active_link(grid, node_values, out=None):
"""Calculate gradients at active links.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 4))
>>> z = np.array([0., 0., 0., 0.,
... 1., 1., 1., 1.,
... 3., 3., 3., 3.])
>>> grid.calc_grad_of_active_link(z)
array([ 1., 1., 0., 0., 0., 2., 2.])
This method is *deprecated*. Instead, use ``calc_grad_at_link``.
>>> vals = grid.calc_grad_at_link(z)
>>> vals[grid.active_links]
array([ 1., 1., 0., 0., 0., 2., 2.])
LLCATS: DEPR
"""
return calc_grad_at_active_link(grid, node_values, out)
@deprecated(use='calc_grad_at_link', version='1.0beta')
@use_field_name_or_array('node')
def calc_grad_at_active_link(grid, node_values, out=None):
"""Calculate gradients of node values over active links.
Calculates the gradient in *quantity* node values at each active link in
the grid.
Construction::
calc_grad_at_active_link(grid, node_values, out=None)
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
Gradients across active links.
LLCATS: DEPR LINF GRAD
"""
if out is None:
out = np.empty(grid.number_of_active_links, dtype=float)
return np.divide(node_values[grid._activelink_tonode] -
node_values[grid._activelink_fromnode],
grid.length_of_link[grid.active_links], out=out)
@deprecated(use='calc_grad_at_link', version='1.0beta')
@use_field_name_or_array('node')
def calculate_gradients_at_faces(grid, node_values, out=None):
"""Calculate gradients of node values over faces.
Calculate and return gradient in *node_values* at each face in the grid.
Gradients are calculated from the nodes at either end of the link that
crosses each face.
Construction::
calculate_gradients_at_faces(grid, node_values, out=None)
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 (x number of faces)
Gradients across faces.
Examples
--------
>>> from landlab import RasterModelGrid
>>> rg = RasterModelGrid(3, 4, 10.0)
>>> z = rg.add_zeros('node', 'topographic__elevation')
>>> z[5] = 50.0
>>> z[6] = 36.0
>>> calculate_gradients_at_faces(rg, z) # there are 7 faces
array([ 5. , 3.6, 5. , -1.4, -3.6, -5. , -3.6])
>>> from landlab import HexModelGrid
>>> hg = HexModelGrid(3, 3, 10.0)
>>> z = hg.add_zeros('node', 'topographic__elevation', noclobber=False)
>>> z[4] = 50.0
>>> z[5] = 36.0
>>> calculate_gradients_at_faces(hg, z) # there are 11 faces
array([ 5. , 5. , 3.6, 3.6, 5. , -1.4, -3.6, -5. , -5. , -3.6, -3.6])
LLCATS: DEPR GRAD
"""
if out is None:
out = grid.empty(at='face')
laf = grid.link_at_face
return np.divide(node_values[grid.node_at_link_head[laf]] -
node_values[grid.node_at_link_tail[laf]],
grid.length_of_link[laf], out=out)
@use_field_name_or_array('node')
def calc_diff_at_link(grid, node_values, out=None):
"""Calculate differences of node values over links.
Calculates the difference in quantity *node_values* at each link in the
grid.
Construction::
calc_diff_at_link(grid, node_values, out=None)
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
>>> rmg = RasterModelGrid((3, 3))
>>> z = np.zeros(9)
>>> z[4] = 1.
>>> rmg.calc_diff_at_link(z)
array([ 0., 0., 0., 1., 0., 1., -1., 0., -1., 0., 0., 0.])
LLCATS: LINF GRAD
"""
if out is None:
out = grid.empty(at='link')
node_values = np.asarray(node_values)
return np.subtract(node_values[grid.node_at_link_head],
node_values[grid.node_at_link_tail], out=out)
@deprecated(use='calc_diff_at_link', version='1.0beta')
@use_field_name_or_array('node')
def calculate_diff_at_links(grid, node_values, out=None):
"""Calculate differences of node values over links.
Examples
--------
>>> import numpy as np
>>> from landlab import RasterModelGrid
>>> grid = RasterModelGrid((3, 3))
>>> z = np.zeros(9)
>>> z[4] = 1.
>>> grid.calculate_diff_at_links(z)
array([ 0., 0., 0., 1., 0., 1., -1., 0., -1., 0., 0., 0.])
>>> grid.calc_diff_at_link(z)
array([ 0., 0., 0., 1., 0., 1., -1., 0., -1., 0., 0., 0.])
LLCATS: DEPR LINF GRAD
"""
return calc_diff_at_link(grid, node_values, out)
@deprecated(use='calc_diff_at_link', version='1.0beta')
@use_field_name_or_array('node')
def calculate_diff_at_active_links(grid, node_values, out=None):
"""Calculate differences of node values over active links.
Calculates the difference in quantity *node_values* at each active link
in the grid.
Construction::
calculate_diff_at_active_links(grid, node_values, out=None)
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 active links.
LLCATS: DEPR LINF GRAD
"""
if out is None:
out = np.empty(grid.number_of_active_links, dtype=float)
node_values = np.asarray(node_values)
return np.subtract(node_values[grid._activelink_tonode],
node_values[grid._activelink_fromnode], out=out)
def calc_unit_normal_at_patch(grid, elevs='topographic__elevation'):
"""Calculate and return the unit normal vector <a, b, c> to a patch.
Parameters
----------
grid : ModelGrid
A ModelGrid.
elevs : str or ndarray, optional
Field name or array of node values.
Returns
-------
nhat : num-patches x length-3 array
The unit normal vector <a, b, c> to each patch.
Examples
--------
>>> from landlab import HexModelGrid
>>> mg = HexModelGrid(3, 3)
>>> z = mg.node_x * 3. / 4.
>>> mg.calc_unit_normal_at_patch(z)
array([[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8],
[-0.6, 0. , 0.8]])
LLCATS: PINF GRAD
"""
try:
z = grid.at_node[elevs]
except TypeError:
z = elevs
# conceptualize patches as sets of 3 nodes, PQR
diff_xyz_PQ = np.empty((grid.number_of_patches, 3))
# ^this is the vector (xQ-xP, yQ-yP, zQ-yP)
diff_xyz_PR = np.empty((grid.number_of_patches, 3))
P = grid.nodes_at_patch[:, 0]
Q = grid.nodes_at_patch[:, 1]
R = grid.nodes_at_patch[:, 2]
x_P = grid.node_x[P]
y_P = grid.node_y[P]
z_P = z[P]
diff_xyz_PQ[:, 0] = grid.node_x[Q] - x_P
diff_xyz_PQ[:, 1] = grid.node_y[Q] - y_P
diff_xyz_PQ[:, 2] = z[Q] - z_P
diff_xyz_PR[:, 0] = grid.node_x[R] - x_P
diff_xyz_PR[:, 1] = grid.node_y[R] - y_P
diff_xyz_PR[:, 2] = z[R] - z_P
# cross product is orthogonal to both vectors, and is the normal
# n = <a, b, c>, where plane is ax + by + cz = d
nhat = np.cross(diff_xyz_PQ, diff_xyz_PR) # <a, b, c>
nmag = np.sqrt(np.square(nhat).sum(axis=1))
return nhat / nmag.reshape(grid.number_of_patches, 1)
def calc_slope_at_patch(grid, elevs='topographic__elevation',
ignore_closed_nodes=True, unit_normal=None):
"""Calculate the slope (positive magnitude of gradient) at patches.
If ignore_closed_nodes is True, closed nodes do not affect slope
calculations. If a closed node is present in a patch, the
patch slope is set to zero.
Parameters
----------
grid : ModelGrid
A ModelGrid.
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.
unit_normal : array with shape (num_patches, 3) (optional)
The unit normal vector to each patch, if already known.
Returns
-------
slopes_at_patch : n_patches-long array
The slope (positive gradient magnitude) of each patch.
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.)
True
LLCATS: PINF GRAD
"""
if unit_normal is not None:
assert unit_normal.shape[1] == 3
nhat = unit_normal
else:
nhat = grid.calc_unit_normal_at_patch(elevs)
dotprod = nhat[:, 2] # by definition
cos_slopes_at_patch = dotprod # ...because it's now a unit vector
slopes_at_patch = np.arccos(cos_slopes_at_patch)
if ignore_closed_nodes:
badnodes = grid.status_at_node[grid.nodes_at_patch] == CLOSED_BOUNDARY
bad_patches = badnodes.sum(axis=1) > 0
slopes_at_patch[bad_patches] = 0.
return slopes_at_patch
def calc_grad_at_patch(grid, elevs='topographic__elevation',
ignore_closed_nodes=True,
unit_normal=None, slope_magnitude=None):
"""Calculate the components of the gradient at each patch.
If ignore_closed_nodes is True, closed nodes do not affect gradient
calculations. If a closed node is present in a patch, the
patch gradient is set to zero in both x and y directions.
Parameters
----------
grid : ModelGrid
A ModelGrid.
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.
unit_normal : array with shape (num_patches, 3) (optional)
The unit normal vector to each patch, if already known.
slope_magnitude : array with size num_patches (optional)
The slope of each patch, if already known.
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 *units*.
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.)
True
>>> np.allclose(x_grad, 0.)
True
LLCATS: PINF GRAD
"""
if unit_normal is not None:
assert unit_normal.shape[1] == 3
nhat = unit_normal
else:
nhat = grid.calc_unit_normal_at_patch(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,
unit_normal=nhat)
theta = np.arctan2(- nhat[:, 1], - nhat[:, 0])
x_slope_patches = np.cos(theta) * slopes_at_patch
y_slope_patches = np.sin(theta) * slopes_at_patch
return (x_slope_patches, y_slope_patches)
def calc_slope_at_node(grid, elevs='topographic__elevation',
method='patch_mean', ignore_closed_nodes=True,
return_components=False, **kwds):
"""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.
It averages the gradient on each of the
patches surrounding the node, creating a value for node slope that
better incorporates nonlocal elevation information. Directional
information can still be returned through use of the return_components
keyword.
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 # not always true
If ignore_closed_nodes is False, all proximal elevation values will be used
in the calculation. If True, only unclosed nodes are used.
Parameters
----------
grid : ModelGrid
A ModelGrid.
elevs : str or ndarray, optional
Field name or array of node values.
method : {'patch_mean', 'Horn'}
By equivalence to the raster version, `'patch_mean'` returns a scalar
mean on the patches; `'Horn'` returns a vector mean on the patches.
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((4, 5), 1.)
>>> z = mg.node_x
>>> slopes = mg.calc_slope_at_node(elevs=z)
>>> np.allclose(slopes, 45. / 180. * np.pi)
True
>>> mg = RasterModelGrid((4, 5), 1.)
>>> z = - mg.node_y
>>> slope_mag, cmp = mg.calc_slope_at_node(elevs=z,
... return_components=True)
>>> np.allclose(slope_mag, np.pi / 4.)
True
>>> np.allclose(cmp[0], 0.)
True
>>> np.allclose(cmp[1], - np.pi / 4.)
True
>>> mg = RadialModelGrid(num_shells=9)
>>> z = mg.radius_at_node
>>> slopes = mg.calc_slope_at_node(elevs=z)
>>> mean_ring_slope = []
>>> for i in range(10):
... mean_ring_slope.append(
... slopes[np.isclose(mg.radius_at_node, i)].mean())
Notice the small amounts of numerical error here:
>>> target_mean_ring_slope = [0.85707194785013108, 0.79363155567711452,
... 0.77922185867135429, 0.78359813570962411,
... 0.78433070957439543, 0.78452745144699965,
... 0.78477643475446901, 0.78506472422668094,
... 0.78505793680521629, 0.78661256633611021]
>>> np.allclose(mean_ring_slope, target_mean_ring_slope)
True
LLCATS: NINF GRAD SURF
"""
if method not in ('patch_mean', 'Horn'):
raise ValueError('method name not understood')
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)
nhat = grid.calc_unit_normal_at_patch(elevs=elevs)
slopes_at_patch = grid.calc_slope_at_patch(
elevs=elevs, ignore_closed_nodes=ignore_closed_nodes, unit_normal=nhat)
# now CAREFUL - patches_at_node is MASKED
slopes_at_node_unmasked = slopes_at_patch[patches_at_node]
slopes_at_node_masked = np.ma.array(slopes_at_node_unmasked,
mask=patches_at_node.mask)
slope_mag = np.mean(slopes_at_node_masked, axis=1).data
if return_components or method == 'Horn':
(x_slope_patches, y_slope_patches) = grid.calc_grad_at_patch(
elevs=elevs, unit_normal=nhat,
ignore_closed_nodes=ignore_closed_nodes,
slope_magnitude=slopes_at_patch)
x_slope_unmasked = x_slope_patches[patches_at_node]
x_slope_masked = np.ma.array(x_slope_unmasked,
mask=patches_at_node.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=patches_at_node.mask)
y_slope = np.mean(y_slope_masked, axis=1).data
mean_grad_x = x_slope
mean_grad_y = y_slope
if method == 'Horn':
slope_mag = np.arctan(np.sqrt(np.tan(y_slope_masked) ** 2 +
np.tan(x_slope_masked) ** 2))
return slope_mag
else:
return slope_mag, (mean_grad_x, mean_grad_y)
else:
return slope_mag
def calc_aspect_at_node(grid, slope_component_tuple=None,
elevs='topographic__elevation', unit='degrees',
ignore_closed_nodes=True):
"""Get array of aspect of a surface.
Calculates at returns the aspect of a surface. Aspect is returned as
radians clockwise of north, unless input parameter units is set to
'degrees'.
If slope_component_tuple is provided, i.e., (slope_x, slope_y), 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.
If ignore_closed_nodes is False, all proximal elevation values will be used
in the calculation. If True, only unclosed nodes are used.
Parameters
----------
grid : ModelGrid
A ModelGrid.
slope_component_tuple : (slope_x_array, slope_y_array) (optional)
Tuple of components of slope in the x and y directions, defined
on nodes, if already known. If not, provide *elevs*.
elevs : str or array (optional)
Node field name or node array of elevations.
If *slope_component_tuple* is not provided, must be set, but unused
otherwise.
unit : {'degrees', 'radians'}
Controls the unit that the aspect is returned as.
ignore_closed_nodes : bool
If True, do not incorporate values at closed nodes into the calc.
Examples
--------
>>> from landlab import RasterModelGrid
>>> mg = RasterModelGrid((4, 4))
>>> z = mg.node_x ** 2 + mg.node_y ** 2
>>> mg.calc_aspect_at_node(elevs=z)
array([ 225. , 240.16585039, 255.2796318 , 258.69006753,
209.83414961, 225. , 243.54632481, 248.77808974,
194.7203682 , 206.45367519, 225. , 231.94498651,
191.30993247, 201.22191026, 218.05501349, 225. ])
>>> z = z.max() - z
>>> mg.calc_aspect_at_node(elevs=z)
array([ 45. , 60.16585039, 75.2796318 , 78.69006753,
29.83414961, 45. , 63.54632481, 68.77808974,
14.7203682 , 26.45367519, 45. , 51.94498651,
11.30993247, 21.22191026, 38.05501349, 45. ])
>>> mg = RasterModelGrid((4, 4), (2., 3.))
>>> z = mg.node_x ** 2 + mg.node_y ** 2
>>> mg.calc_aspect_at_node(elevs=z)
array([ 236.30993247, 247.52001262, 259.97326008, 262.40535663,
220.75264634, 234.41577266, 251.13402374, 255.29210302,
201.54258265, 215.47930877, 235.73541937, 242.24162456,
196.69924423, 209.43534223, 229.19345757, 236.30993247])
Note that a small amount of asymmetry arises at the grid edges due
to the "missing" nodes beyond the edge of the grid.
LLCATS: NINF SURF
"""
if slope_component_tuple:
if not isinstance(slope_component_tuple, (tuple, list)):
raise TypeError('slope_component_tuple must be tuple')
if len(slope_component_tuple) != 2:
raise ValueError('slope_component_tuple must be of length 2')
else:
try:
elev_array = grid.at_node[elevs]
except (KeyError, TypeError):
assert elevs.size == grid.number_of_nodes
elev_array = elevs
_, slope_component_tuple = grid.calc_slope_at_node(
elevs=elev_array, ignore_closed_nodes=ignore_closed_nodes,
return_components=True)
angle_from_x_ccw = np.arctan2(
- slope_component_tuple[1], - slope_component_tuple[0])
if unit == 'degrees':
return radians_to_degrees(angle_from_x_ccw)
elif unit == 'radians':
angle_from_north_cw = (5. * np.pi / 2. -
angle_from_x_ccw) % (2. * np.pi)
return angle_from_north_cw
else:
raise TypeError("unit must be 'degrees' or 'radians'")
