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Example 5 - stripy smoothing operations

Example 5 - stripy smoothing operations

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.

Here we demonstrate how to access SSRFPACK smoothing through the stripy interface

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 with noise and short wavelengths

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_noisy(lons, lats, k1, k2, noise, short):
     return  np.cos(k1*lons) * np.sin(k2*lats) + short * (np.cos(k1*5.0*lons) * np.sin(k2*5.0*lats)) +  noise * np.random.random(lons.shape)

# 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_n = analytic_noisy(mesh.lons, mesh.lats, 5.0, 2.0, 0.1, 0.0)

%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="#999999", linewidth=2.0)
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_n-analytic_sol, cmap=plt.cm.RdBu)

pass

Smoothing operations

The sTriangulation.smoothing method directly wraps the SSRFPack smoother that smooths a surface f described by values on the mesh vertices to find a new surface f’ (also described on the mesh vertices) by choosing nodal function values and gradients to minimize the linearized curvature of F subject to a bound on the deviation from the data values.


help(mesh.smoothing)

smoothing(self, f, w, sm, smtol, gstol)

method of stripy.spherical_meshes.icosahedral_mesh instance
Smooths a surface f by choosing nodal function values and gradients to
minimize the linearized curvature of F subject to a bound on the
deviation from the data values. This is more appropriate than interpolation
when significant errors are present in the data.

Parameters
----------
 f : array of floats, shape (n,)
    field to apply smoothing on
 w : array of floats, shape (n,)
    weights associated with data value in f
    w[i] = 1/sigma_f^2 is a good rule of thumb.
 sm : float
    positive parameter specifying an upper bound on Q2(f).
    generally n-sqrt(2n) <= sm <= n+sqrt(2n)
 smtol : float
    specifies relative error in satisfying the constraint
    sm(1-smtol) <= Q2 <= sm(1+smtol) between 0 and 1.
 gstol : float
    tolerance for convergence.
    gstol = 0.05*mean(sigma_f)^2 is a good rule of thumb.

Returns
-------
 f_smooth : array of floats, shape (n,)
    smoothed version of f
 (dfdx, dfdy, dfdz) : tuple of floats, tuple of 3 shape (n,) arrays
    first derivatives of f_smooth in the x, y, and z directions
 err : error indicator

stripy_smoothed,  dds, err = mesh.smoothing(analytic_sol_n, np.ones_like(analytic_sol_n), 10.0, 0.1, 0.01)
stripy_smoothed2, dds, err = mesh.smoothing(analytic_sol_n, np.ones_like(analytic_sol_n), 1.0, 0.1, 0.01)
stripy_smoothed3, dds, err = mesh.smoothing(analytic_sol_n, np.ones_like(analytic_sol_n), 50.0, 0.1, 0.01)

delta_n  = analytic_sol_n - stripy_smoothed
delta_ns = analytic_sol   - stripy_smoothed

delta_n2  = analytic_sol_n - stripy_smoothed2
delta_ns2 = analytic_sol   - stripy_smoothed2

delta_n3  = analytic_sol_n - stripy_smoothed3
delta_ns3 = analytic_sol   - stripy_smoothed3

Results of smoothing with different value of sm

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=stripy_smoothed,
                   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 = { "unsmoothed": analytic_sol_n,
             "smooth1": stripy_smoothed, 
             "smooth2": stripy_smoothed2, 
             "smooth3": stripy_smoothed3,              
             "Delta smooth1": analytic_sol_n - stripy_smoothed, 
             "Delta smooth2": analytic_sol_n - stripy_smoothed2, 
             "Delta smooth3": analytic_sol_n - stripy_smoothed3, }

@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 Ex6-Scattered-Data

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Example 4 - stripy gradients on the sphere

next

Example 6 - Scattered Data and ‘Heat Maps’