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Example 4 - stripy gradients on the sphere

Example 4 - stripy gradients on the sphere

SSRFPACK is a Fortran 77 software package that constructs a smooth interpolatory or approximating surface to data values associated with arbitrarily distributed points on the surface of a sphere. It employs automatically selected tension factors to preserve shape properties of the data and avoid overshoot and undershoot associated with steep gradients.

Define a computational mesh

Use the (usual) icosahedron with face points included.

import stripy as stripy

mesh = stripy.spherical_meshes.icosahedral_mesh(refinement_levels=4, include_face_points=True)

print(mesh.npoints)

Analytic function

Define a relatively smooth function that we can interpolate from the coarse mesh to the fine mesh and analyse

import numpy as np

def analytic(lons, lats, k1, k2):
     return np.cos(k1*lons) * np.sin(k2*lats)

def analytic_ddlon(lons, lats, k1, k2):
     return -k1 * np.sin(k1*lons) * np.sin(k2*lats) / np.cos(lats)

def analytic_ddlat(lons, lats, k1, k2):
     return k2 * np.cos(k1*lons) * np.cos(k2*lats) 

analytic_sol = analytic(mesh.lons, mesh.lats, 5.0, 2.0)
analytic_sol_ddlon = analytic_ddlon(mesh.lons, mesh.lats, 5.0, 2.0)
analytic_sol_ddlat = analytic_ddlat(mesh.lons, mesh.lats, 5.0, 2.0)

%matplotlib inline

import cartopy
import cartopy.crs as ccrs
import matplotlib.pyplot as plt


fig = plt.figure(figsize=(10, 10), facecolor="none")
ax  = plt.subplot(111, projection=ccrs.Orthographic(central_longitude=0.0, central_latitude=0.0, globe=None))
ax.coastlines(color="lightgrey")
ax.set_global()

lons0 = np.degrees(mesh.lons)
lats0 = np.degrees(mesh.lats)

ax.scatter(lons0, lats0, 
            marker="o", s=10.0, transform=ccrs.PlateCarree(), c=analytic_sol, cmap=plt.cm.RdBu)

pass

Derivatives of solution compared to analytic values

The gradient_lonlat method of the sTriangulation takes a data array reprenting values on the mesh vertices and returns the lon and lat derivatives. There is an equivalent gradient_xyz method which returns the raw derivatives in Cartesian form. Although this is generally less useful, if you are computing the slope (for example) that can be computed in either coordinate system and may cross the pole, consider using the Cartesian form.

stripy_ddlon, stripy_ddlat = mesh.gradient_lonlat(analytic_sol)

import k3d
plot = k3d.plot(camera_auto_fit=False, grid_visible=False, 
                menu_visibility=True, axes_helper=False )

indices = mesh.simplices.astype(np.uint32)
points = np.column_stack(mesh.points.T).astype(np.float32)

mesh_viewer = k3d.mesh(points, indices, wireframe=False, attribute=analytic_sol,
                   color_map=k3d.colormaps.basic_color_maps.CoolWarm, 
                   name="original",
                   flat_shading=False, opacity=1.0  )

plot   += mesh_viewer
plot   += k3d.points(points, point_size=0.01,color=0x779977)


plot.display()

## ## ## 

from ipywidgets import interact, interactive
import ipywidgets as widgets

choices = { "analytic": analytic_sol,
             "stripy ddlon": stripy_ddlon, 
             "stripy ddlat": stripy_ddlat, 
             "error ddlon":  stripy_ddlon-analytic_sol_ddlon, 
             "error ddlat":  stripy_ddlat-analytic_sol_ddlat }

@interact(choice=choices.keys())
def chooser(choice):
    mesh_viewer.attribute = choices[choice].astype(np.float32)
    range = np.sqrt((choices[choice]**2).mean()) * 0.5
    mesh_viewer.color_range = [-range, range]
    return

The next example is Ex5-Smoothing

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Example 3 - stripy interpolation on the sphere

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Example 5 - stripy smoothing operations