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# Copyright (c) 2008-2016 MetPy Developers. |
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# Distributed under the terms of the BSD 3-Clause License. |
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# SPDX-License-Identifier: BSD-3-Clause |
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""" |
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Inverse Distance Verification: Cressman and Barnes |
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================================================== |
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Compare inverse distance interpolation methods |
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Two popular interpolation schemes that use inverse distance weighting of observations are the |
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Barnes and Cressman analyses. The Cressman analysis is relatively straightforward and uses |
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the ratio between distance of an observation from a grid cell and the maximum allowable |
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distance to calculate the relative importance of an observation for calculating an |
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interpolation value. Barnes uses the inverse exponential ratio of each distance between |
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an observation and a grid cell and the average spacing of the observations over the domain. |
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Algorithmically: |
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1. A KDTree data structure is built using the locations of each observation. |
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2. All observations within a maximum allowable distance of a particular grid cell are found in |
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O(log n) time. |
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3. Using the weighting rules for Cressman or Barnes analyses, the observations are given a |
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proportional value, primarily based on their distance from the grid cell. |
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4. The sum of these proportional values is calculated and this value is used as the |
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interpolated value. |
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5. Steps 2 through 4 are repeated for each grid cell. |
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""" |
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import matplotlib.pyplot as plt |
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import numpy as np |
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from scipy.spatial import cKDTree |
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from scipy.spatial.distance import cdist |
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from metpy.gridding.gridding_functions import calc_kappa |
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from metpy.gridding.interpolation import barnes_point, cressman_point |
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from metpy.gridding.triangles import dist_2 |
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plt.rcParams['figure.figsize'] = (15, 10) |
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def draw_circle(x, y, r, m, label): |
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nx = x + r * np.cos(np.deg2rad(list(range(360)))) |
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ny = y + r * np.sin(np.deg2rad(list(range(360)))) |
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plt.plot(nx, ny, m, label=label) |
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########################################### |
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# Generate random x and y coordinates, and observation values proportional to x * y. |
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# |
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# Set up two test grid locations at (30, 30) and (60, 60). |
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np.random.seed(100) |
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pts = np.random.randint(0, 100, (10, 2)) |
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xp = pts[:, 0] |
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yp = pts[:, 1] |
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zp = xp * xp / 1000 |
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sim_gridx = [30, 60] |
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sim_gridy = [30, 60] |
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########################################### |
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# Set up a cKDTree object and query all of the observations within "radius" of each grid point. |
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# |
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# The variable ``indices`` represents the index of each matched coordinate within the |
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# cKDTree's ``data`` list. |
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grid_points = np.array(list(zip(sim_gridx, sim_gridy))) |
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radius = 40 |
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obs_tree = cKDTree(list(zip(xp, yp))) |
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indices = obs_tree.query_ball_point(grid_points, r=radius) |
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########################################### |
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# For grid 0, we will use Cressman to interpolate its value. |
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x1, y1 = obs_tree.data[indices[0]].T |
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cress_dist = dist_2(sim_gridx[0], sim_gridy[0], x1, y1) |
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cress_obs = zp[indices[0]] |
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cress_val = cressman_point(cress_dist, cress_obs, radius) |
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########################################### |
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# For grid 1, we will use barnes to interpolate its value. |
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# |
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# We need to calculate kappa--the average distance between observations over the domain. |
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x2, y2 = obs_tree.data[indices[1]].T |
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barnes_dist = dist_2(sim_gridx[1], sim_gridy[1], x2, y2) |
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barnes_obs = zp[indices[1]] |
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ave_spacing = np.mean((cdist(list(zip(xp, yp)), list(zip(xp, yp))))) |
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kappa = calc_kappa(ave_spacing) |
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barnes_val = barnes_point(barnes_dist, barnes_obs, kappa) |
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########################################### |
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# Plot all of the affiliated information and interpolation values. |
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for i, zval in enumerate(zp): |
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plt.plot(pts[i, 0], pts[i, 1], '.') |
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plt.annotate(str(zval) + ' F', xy=(pts[i, 0] + 2, pts[i, 1])) |
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plt.plot(sim_gridx, sim_gridy, '+', markersize=10) |
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plt.plot(x1, y1, 'ko', fillstyle='none', markersize=10, label='grid 0 matches') |
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plt.plot(x2, y2, 'ks', fillstyle='none', markersize=10, label='grid 1 matches') |
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draw_circle(sim_gridx[0], sim_gridy[0], m='k-', r=radius, label='grid 0 radius') |
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draw_circle(sim_gridx[1], sim_gridy[1], m='b-', r=radius, label='grid 1 radius') |
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plt.annotate('grid 0: cressman {:.3f}'.format(cress_val), xy=(sim_gridx[0] + 2, sim_gridy[0])) |
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plt.annotate('grid 1: barnes {:.3f}'.format(barnes_val), xy=(sim_gridx[1] + 2, sim_gridy[1])) |
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plt.axes().set_aspect('equal', 'datalim') |
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plt.legend() |
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########################################### |
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# For each point, we will do a manual check of the interpolation values by doing a step by |
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# step and visual breakdown. |
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# |
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# Plot the grid point, observations within radius of the grid point, their locations, and |
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# their distances from the grid point. |
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plt.annotate('grid 0: ({}, {})'.format(sim_gridx[0], sim_gridy[0]), |
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xy=(sim_gridx[0] + 2, sim_gridy[0])) |
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plt.plot(sim_gridx[0], sim_gridy[0], '+', markersize=10) |
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mx, my = obs_tree.data[indices[0]].T |
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mz = zp[indices[0]] |
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for x, y, z in zip(mx, my, mz): |
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d = np.sqrt((sim_gridx[0] - x)**2 + (y - sim_gridy[0])**2) |
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plt.plot([sim_gridx[0], x], [sim_gridy[0], y], '--') |
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xave = np.mean([sim_gridx[0], x]) |
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yave = np.mean([sim_gridy[0], y]) |
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plt.annotate('distance: {}'.format(d), xy=(xave, yave)) |
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plt.annotate('({}, {}) : {} F'.format(x, y, z), xy=(x, y)) |
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plt.xlim(0, 80) |
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plt.ylim(0, 80) |
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plt.axes().set_aspect('equal', 'datalim') |
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########################################### |
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# Step through the cressman calculations. |
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dists = np.array([22.803508502, 7.21110255093, 31.304951685, 33.5410196625]) |
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values = np.array([0.064, 1.156, 3.364, 0.225]) |
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cres_weights = (radius * radius - dists * dists) / (radius * radius + dists * dists) |
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total_weights = np.sum(cres_weights) |
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proportion = cres_weights / total_weights |
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value = values * proportion |
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val = cressman_point(cress_dist, cress_obs, radius) |
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print('Manual cressman value for grid 1:\t', np.sum(value)) |
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print('Metpy cressman value for grid 1:\t', val) |
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########################################### |
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# Now repeat for grid 1, except use barnes interpolation. |
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plt.annotate('grid 1: ({}, {})'.format(sim_gridx[1], sim_gridy[1]), |
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xy=(sim_gridx[1] + 2, sim_gridy[1])) |
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plt.plot(sim_gridx[1], sim_gridy[1], '+', markersize=10) |
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mx, my = obs_tree.data[indices[1]].T |
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mz = zp[indices[1]] |
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for x, y, z in zip(mx, my, mz): |
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d = np.sqrt((sim_gridx[1] - x)**2 + (y - sim_gridy[1])**2) |
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plt.plot([sim_gridx[1], x], [sim_gridy[1], y], '--') |
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xave = np.mean([sim_gridx[1], x]) |
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yave = np.mean([sim_gridy[1], y]) |
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plt.annotate('distance: {}'.format(d), xy=(xave, yave)) |
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plt.annotate('({}, {}) : {} F'.format(x, y, z), xy=(x, y)) |
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plt.xlim(40, 80) |
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plt.ylim(40, 100) |
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plt.axes().set_aspect('equal', 'datalim') |
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########################################### |
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# Step through barnes calculations. |
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dists = np.array([9.21954445729, 22.4722050542, 27.892651362, 38.8329756779]) |
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values = np.array([2.809, 6.241, 4.489, 2.704]) |
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weights = np.exp(-dists**2 / kappa) |
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total_weights = np.sum(weights) |
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value = np.sum(values * (weights / total_weights)) |
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print('Manual barnes value:\t', value) |
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print('Metpy barnes value:\t', barnes_point(barnes_dist, barnes_obs, kappa)) |
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Unidata /
MetPy
