Example 6 - Scattered Data and ‘Heat Maps’#
There are different ways to map point data to a smooth field. One way is to triangulate the data, smooth it and interpolate to a regular mesh (see previous notebooks). It is also possible to construct weighted averages from scattered points to a regular mesh. In this notebook we work through how to find where points lie in the mesh and map their values to nearby vertices.
Notebook contents#
The next example is Ex7-Refinement-of-Triangulations
Point data with uneven spatial distribution#
Define a relatively smooth function that we can interpolate from the coarse mesh to the fine mesh and analyse. At local scales it is convenient to use projected coordinate reference systems (CRS) to work in metres instead of degrees. We use the heat flow database for Southeastern Australia from Mather et al. 2017
import numpy as np
HFdata = np.loadtxt("../Data/HeatFlowSEAustralia.csv", delimiter=',', usecols=(3,4,5), skiprows=1)
eastings = HFdata[:,0]
northings = HFdata[:,1]
heat_flow = HFdata[:,2]
%matplotlib inline
import matplotlib.pyplot as plt
import cartopy
import cartopy.crs as ccrs
# local coordinate reference system
proj = ccrs.epsg(28354)
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
ax.coastlines(resolution='10m')
ax.set_extent([135, 147, -39, -30])
ax.scatter(eastings, northings,
marker="o", cmap=plt.cm.RdBu, c=heat_flow, transform=proj)
ax.gridlines(draw_labels=True)
plt.show()
---------------------------------------------------------------------------
ModuleNotFoundError Traceback (most recent call last)
Cell In[2], line 1
----> 1 get_ipython().run_line_magic('matplotlib', 'inline')
2 import matplotlib.pyplot as plt
3 import cartopy
File ~/.local/lib/python3.12/site-packages/IPython/core/interactiveshell.py:2482, in InteractiveShell.run_line_magic(self, magic_name, line, _stack_depth)
2480 kwargs['local_ns'] = self.get_local_scope(stack_depth)
2481 with self.builtin_trap:
-> 2482 result = fn(*args, **kwargs)
2484 # The code below prevents the output from being displayed
2485 # when using magics with decorator @output_can_be_silenced
2486 # when the last Python token in the expression is a ';'.
2487 if getattr(fn, magic.MAGIC_OUTPUT_CAN_BE_SILENCED, False):
File ~/.local/lib/python3.12/site-packages/IPython/core/magics/pylab.py:103, in PylabMagics.matplotlib(self, line)
98 print(
99 "Available matplotlib backends: %s"
100 % _list_matplotlib_backends_and_gui_loops()
101 )
102 else:
--> 103 gui, backend = self.shell.enable_matplotlib(args.gui)
104 self._show_matplotlib_backend(args.gui, backend)
File ~/.local/lib/python3.12/site-packages/IPython/core/interactiveshell.py:3667, in InteractiveShell.enable_matplotlib(self, gui)
3664 import matplotlib_inline.backend_inline
3666 from IPython.core import pylabtools as pt
-> 3667 gui, backend = pt.find_gui_and_backend(gui, self.pylab_gui_select)
3669 if gui != None:
3670 # If we have our first gui selection, store it
3671 if self.pylab_gui_select is None:
File ~/.local/lib/python3.12/site-packages/IPython/core/pylabtools.py:338, in find_gui_and_backend(gui, gui_select)
321 def find_gui_and_backend(gui=None, gui_select=None):
322 """Given a gui string return the gui and mpl backend.
323
324 Parameters
(...)
335 'WXAgg','Qt4Agg','module://matplotlib_inline.backend_inline','agg').
336 """
--> 338 import matplotlib
340 if _matplotlib_manages_backends():
341 backend_registry = matplotlib.backends.registry.backend_registry
ModuleNotFoundError: No module named 'matplotlib'
Define a regular computational mesh#
Use the (usual) icosahedron with face points included.
import stripy as stripy
xmin = eastings.min()
xmax = eastings.max()
ymin = northings.min()
ymax = northings.max()
extent = [xmin, xmax, ymin, ymax]
# define a mesh with 20km x 20km resolution
spacingX = 10000.0
spacingY = 10000.0
mesh = stripy.cartesian_meshes.square_mesh(extent, spacingX, spacingY, refinement_levels=0, tree=True)
print("number of points = {}".format(mesh.npoints))
number of points = 45920
Count heat flow points per triangle#
This is a numpy wrapper around the TRIPACK routine which operates by retriangulation and is therefore not particularly fast.
triangles = mesh.containing_triangle(eastings, northings)
## The return_counts argument is quite new in numpy
try:
tris, counts = np.unique(triangles, return_counts=True)
except:
def unique_count(a):
unique, inverse = np.unique(a, return_inverse=True)
count = np.zeros(len(unique), np.int)
np.add.at(count, inverse, 1)
return unique, count
tris, counts = unique_count(triangles)
print("number of unique triangles = {}".format(tris.shape[0]))
number of unique triangles = 270
## map to nodes so we can plot this
hit_count = np.zeros(mesh.npoints)
for i in range(0, tris.shape[0]):
hit_count[mesh.simplices[tris[i]]] += counts[i]
hit_count /= 3.0
print("mean number of hits = {}".format(hit_count.mean()))
mean number of hits = 0.006402439024390243
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
ax.coastlines(resolution='10m')
ax.set_extent([135, 147, -39, -30])
ax.scatter(mesh.x, mesh.y,
marker="o", cmap=plt.cm.Reds, s=100, c=hit_count, alpha=0.33, transform=proj)
ax.gridlines(draw_labels=True)
plt.show()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[7], line 1
----> 1 fig = plt.figure(figsize=(10,5))
2 ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
3 ax.coastlines(resolution='10m')
NameError: name 'plt' is not defined
Count points per vertex#
The Triangulation.nearest_vertices method uses a k-d tree to find the nearest vertices to a set of x,y points. It returns the nearest vertices and euclidean distance. This requires the k-d tree to have been built when the mesh was initialised (tree=True)
distances, vertices = mesh.nearest_vertices(eastings, northings, k=1)
nodes, ncounts = np.unique(vertices, return_counts=True)
hit_countn = np.zeros(mesh.npoints)
hit_countn[nodes] = ncounts
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
ax.coastlines(resolution='10m')
ax.set_extent([135, 147, -39, -30])
ax.scatter(mesh.x, mesh.y,
marker="o", cmap=plt.cm.Reds, s=100, c=hit_countn, alpha=0.33, transform=proj)
ax.gridlines(draw_labels=True)
plt.show()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[9], line 1
----> 1 fig = plt.figure(figsize=(10,5))
2 ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
3 ax.coastlines(resolution='10m')
NameError: name 'plt' is not defined
Inverse distance weighted number of measurement points#
The k-d tree method provides a specified number of neighbours and the distance to those neighbours. This can be used in a number of ways to smooth or amalgamate data. Here for example is a weighted average of each earthquake to nearby nodes.
We compute the distances to $N$ nearby vertices and distribute information to those vertices in inverse proportion to their distance.
$$ w _i = \frac{d i}{\sum{i=1}^N d _i} $$
Alternatively, we might map information to the vertices by applying a radially symmetric kernel to the point data without normalising.
distances, vertices = mesh.nearest_vertices(eastings, northings, k=100)
norm = distances.sum(axis=1)
# distances, vertices are arrays of shape (data_size, 10)
hit_countid = np.zeros(mesh.npoints)
## numpy shouldn't try to vectorise this reduction operation
for i in range(0,distances.shape[0]):
hit_countid[vertices[i,:]] += distances[i,:] / norm[i]
hit_countidr = np.zeros(mesh.npoints)
## numpy shouldn't try to vectorise this reduction operation
for i in range(0,distances.shape[0]):
hit_countidr[vertices[i,:]] += np.exp( -distances[i,:] / 0.02 )
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
ax.coastlines(resolution='10m')
ax.set_extent([135, 147, -39, -30])
ax.scatter(mesh.x, mesh.y,
marker="o", cmap=plt.cm.Reds, s=100, c=hit_countid, alpha=0.33, transform=proj)
ax.gridlines(draw_labels=True)
plt.show()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[11], line 1
----> 1 fig = plt.figure(figsize=(10,5))
2 ax = fig.add_subplot(111, projection=ccrs.PlateCarree())
3 ax.coastlines(resolution='10m')
NameError: name 'plt' is not defined
The next example is Ex7-Refinement-of-Triangulations