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Table of Contents

• 1  Compare weighted and unweighted mean temperature

  • • 1.0.1  Data

    • 1.0.2  Creating weights

    • 1.0.3  Weighted mean

    • 1.0.4  Plot: comparison with unweighted mean

Compare weighted and unweighted mean temperature

Author: Mathias Hauser

We use the air_temperature example dataset to calculate the area-weighted temperature over its domain. This dataset has a regular latitude/ longitude grid, thus the grid cell area decreases towards the pole. For this grid we can use the cosine of the latitude as proxy for the grid cell area.

%matplotlib inline

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

import xarray as xr

Data

Load the data, convert to celsius, and resample to daily values

ds = xr.tutorial.load_dataset("air_temperature")

# to celsius
air = ds.air - 273.15

# resample from 6-hourly to daily values
air = air.resample(time="D").mean()

air

<xarray.DataArray 'air' (time: 730, lat: 25, lon: 53)>
array([[[-31.2775   , -30.849998 , -30.475002 , ..., -39.7775   ,
         -37.975    , -35.475002 ],
        [-28.575005 , -28.5775   , -28.874996 , ..., -41.9025   ,
         -40.324997 , -36.85     ],
        [-19.149998 , -19.927498 , -21.3275   , ..., -41.675    ,
         -39.454998 , -34.524998 ],
        ...,
        [ 23.15001  ,  22.824997 ,  22.849998 , ...,  22.747505 ,
          22.170013 ,  21.795006 ],
        [ 23.174995 ,  23.574997 ,  23.592514 , ...,  23.022507 ,
          22.850006 ,  22.397507 ],
        [ 23.470009 ,  23.845001 ,  23.950005 , ...,  23.872505 ,
          23.897507 ,  23.82251  ]],

       [[-29.550003 , -29.650005 , -29.849998 , ..., -34.177498 ,
         -32.3525   , -30.0775   ],
        [-25.3275   , -25.95     , -26.927498 , ..., -37.225    ,
         -36.552498 , -34.550003 ],
        [-19.627502 , -21.0775   , -22.852497 , ..., -35.452496 ,
         -34.277496 , -31.25     ],
...
        [ 23.215004 ,  22.265    ,  22.015007 , ...,  23.740005 ,
          23.195007 ,  22.195    ],
        [ 24.3675   ,  24.514992 ,  23.895012 , ...,  23.415    ,
          22.995003 ,  22.269997 ],
        [ 25.417496 ,  25.592499 ,  25.192497 , ...,  23.642502 ,
          23.190002 ,  22.720001 ]],

       [[-28.935001 , -29.535    , -30.385002 , ..., -29.410004 ,
         -28.960003 , -28.46     ],
        [-23.834995 , -24.060001 , -24.559998 , ..., -32.585    ,
         -31.635002 , -30.035004 ],
        [-10.209999 , -10.784988 , -11.434998 , ..., -33.684998 ,
         -31.035    , -27.135002 ],
        ...,
        [ 21.69001  ,  21.990005 ,  23.489998 , ...,  22.265007 ,
          22.015    ,  21.415009 ],
        [ 23.390007 ,  24.439995 ,  24.94001  , ...,  22.415009 ,
          22.315002 ,  21.640007 ],
        [ 24.840012 ,  25.590004 ,  25.54     , ...,  23.065002 ,
          22.715004 ,  22.390007 ]]], dtype=float32)
Coordinates:
  * lat      (lat) float32 75.0 72.5 70.0 67.5 65.0 ... 25.0 22.5 20.0 17.5 15.0
  * lon      (lon) float32 200.0 202.5 205.0 207.5 ... 322.5 325.0 327.5 330.0
  * time     (time) datetime64[ns] 2013-01-01 2013-01-02 ... 2014-12-31

xarray.DataArray

'air'

• time: 730
• lat: 25
• lon: 53

• -31.28 -30.85 -30.48 -30.15 -29.85 ... 24.01 23.89 23.07 22.72 22.39

  array([[[-31.2775   , -30.849998 , -30.475002 , ..., -39.7775   ,
           -37.975    , -35.475002 ],
          [-28.575005 , -28.5775   , -28.874996 , ..., -41.9025   ,
           -40.324997 , -36.85     ],
          [-19.149998 , -19.927498 , -21.3275   , ..., -41.675    ,
           -39.454998 , -34.524998 ],
          ...,
          [ 23.15001  ,  22.824997 ,  22.849998 , ...,  22.747505 ,
            22.170013 ,  21.795006 ],
          [ 23.174995 ,  23.574997 ,  23.592514 , ...,  23.022507 ,
            22.850006 ,  22.397507 ],
          [ 23.470009 ,  23.845001 ,  23.950005 , ...,  23.872505 ,
            23.897507 ,  23.82251  ]],
  
         [[-29.550003 , -29.650005 , -29.849998 , ..., -34.177498 ,
           -32.3525   , -30.0775   ],
          [-25.3275   , -25.95     , -26.927498 , ..., -37.225    ,
           -36.552498 , -34.550003 ],
          [-19.627502 , -21.0775   , -22.852497 , ..., -35.452496 ,
           -34.277496 , -31.25     ],
  ...
          [ 23.215004 ,  22.265    ,  22.015007 , ...,  23.740005 ,
            23.195007 ,  22.195    ],
          [ 24.3675   ,  24.514992 ,  23.895012 , ...,  23.415    ,
            22.995003 ,  22.269997 ],
          [ 25.417496 ,  25.592499 ,  25.192497 , ...,  23.642502 ,
            23.190002 ,  22.720001 ]],
  
         [[-28.935001 , -29.535    , -30.385002 , ..., -29.410004 ,
           -28.960003 , -28.46     ],
          [-23.834995 , -24.060001 , -24.559998 , ..., -32.585    ,
           -31.635002 , -30.035004 ],
          [-10.209999 , -10.784988 , -11.434998 , ..., -33.684998 ,
           -31.035    , -27.135002 ],
          ...,
          [ 21.69001  ,  21.990005 ,  23.489998 , ...,  22.265007 ,
            22.015    ,  21.415009 ],
          [ 23.390007 ,  24.439995 ,  24.94001  , ...,  22.415009 ,
            22.315002 ,  21.640007 ],
          [ 24.840012 ,  25.590004 ,  25.54     , ...,  23.065002 ,
            22.715004 ,  22.390007 ]]], dtype=float32)

• Coordinates: (3)
  • lat
    (lat)
    float32
    75.0 72.5 70.0 ... 20.0 17.5 15.0

    standard_name :

    latitude

    long_name :

    Latitude

    units :

    degrees_north

    axis :

    Y

    array([75. , 72.5, 70. , 67.5, 65. , 62.5, 60. , 57.5, 55. , 52.5, 50. , 47.5,
           45. , 42.5, 40. , 37.5, 35. , 32.5, 30. , 27.5, 25. , 22.5, 20. , 17.5,
           15. ], dtype=float32)

  • lon
    (lon)
    float32
    200.0 202.5 205.0 ... 327.5 330.0

    standard_name :

    longitude

    long_name :

    Longitude

    units :

    degrees_east

    axis :

    X

    array([200. , 202.5, 205. , 207.5, 210. , 212.5, 215. , 217.5, 220. , 222.5,
           225. , 227.5, 230. , 232.5, 235. , 237.5, 240. , 242.5, 245. , 247.5,
           250. , 252.5, 255. , 257.5, 260. , 262.5, 265. , 267.5, 270. , 272.5,
           275. , 277.5, 280. , 282.5, 285. , 287.5, 290. , 292.5, 295. , 297.5,
           300. , 302.5, 305. , 307.5, 310. , 312.5, 315. , 317.5, 320. , 322.5,
           325. , 327.5, 330. ], dtype=float32)

  • time
    (time)
    datetime64[ns]
    2013-01-01 ... 2014-12-31

    array(['2013-01-01T00:00:00.000000000', '2013-01-02T00:00:00.000000000',
           '2013-01-03T00:00:00.000000000', ..., '2014-12-29T00:00:00.000000000',
           '2014-12-30T00:00:00.000000000', '2014-12-31T00:00:00.000000000'],
          dtype='datetime64[ns]')

• Indexes: (3)
  • lat
    PandasIndex

    PandasIndex(Index([75.0, 72.5, 70.0, 67.5, 65.0, 62.5, 60.0, 57.5, 55.0, 52.5, 50.0, 47.5,
           45.0, 42.5, 40.0, 37.5, 35.0, 32.5, 30.0, 27.5, 25.0, 22.5, 20.0, 17.5,
           15.0],
          dtype='float32', name='lat'))

  • lon
    PandasIndex

    PandasIndex(Index([200.0, 202.5, 205.0, 207.5, 210.0, 212.5, 215.0, 217.5, 220.0, 222.5,
           225.0, 227.5, 230.0, 232.5, 235.0, 237.5, 240.0, 242.5, 245.0, 247.5,
           250.0, 252.5, 255.0, 257.5, 260.0, 262.5, 265.0, 267.5, 270.0, 272.5,
           275.0, 277.5, 280.0, 282.5, 285.0, 287.5, 290.0, 292.5, 295.0, 297.5,
           300.0, 302.5, 305.0, 307.5, 310.0, 312.5, 315.0, 317.5, 320.0, 322.5,
           325.0, 327.5, 330.0],
          dtype='float32', name='lon'))

  • time
    PandasIndex

    PandasIndex(DatetimeIndex(['2013-01-01', '2013-01-02', '2013-01-03', '2013-01-04',
                   '2013-01-05', '2013-01-06', '2013-01-07', '2013-01-08',
                   '2013-01-09', '2013-01-10',
                   ...
                   '2014-12-22', '2014-12-23', '2014-12-24', '2014-12-25',
                   '2014-12-26', '2014-12-27', '2014-12-28', '2014-12-29',
                   '2014-12-30', '2014-12-31'],
                  dtype='datetime64[ns]', name='time', length=730, freq='D'))

• Attributes: (0)

Plot the first timestep:

projection = ccrs.LambertConformal(central_longitude=-95, central_latitude=45)

f, ax = plt.subplots(subplot_kw=dict(projection=projection))

air.isel(time=0).plot(transform=ccrs.PlateCarree(), cbar_kwargs=dict(shrink=0.7))
ax.coastlines()

<cartopy.mpl.feature_artist.FeatureArtist at 0x7f39dc3f07f0>

Creating weights

For a rectangular grid the cosine of the latitude is proportional to the grid cell area.

weights = np.cos(np.deg2rad(air.lat))
weights.name = "weights"
weights

<xarray.DataArray 'weights' (lat: 25)>
array([0.25881907, 0.30070582, 0.34202015, 0.38268346, 0.42261827,
       0.4617486 , 0.49999997, 0.5372996 , 0.57357645, 0.6087614 ,
       0.6427876 , 0.67559016, 0.70710677, 0.7372773 , 0.76604444,
       0.7933533 , 0.81915206, 0.8433914 , 0.8660254 , 0.8870108 ,
       0.90630776, 0.9238795 , 0.9396926 , 0.95371693, 0.9659258 ],
      dtype=float32)
Coordinates:
  * lat      (lat) float32 75.0 72.5 70.0 67.5 65.0 ... 25.0 22.5 20.0 17.5 15.0
Attributes:
    standard_name:  latitude
    long_name:      Latitude
    units:          degrees_north
    axis:           Y

xarray.DataArray

'weights'

• lat: 25

• 0.2588 0.3007 0.342 0.3827 0.4226 ... 0.9239 0.9397 0.9537 0.9659

  array([0.25881907, 0.30070582, 0.34202015, 0.38268346, 0.42261827,
         0.4617486 , 0.49999997, 0.5372996 , 0.57357645, 0.6087614 ,
         0.6427876 , 0.67559016, 0.70710677, 0.7372773 , 0.76604444,
         0.7933533 , 0.81915206, 0.8433914 , 0.8660254 , 0.8870108 ,
         0.90630776, 0.9238795 , 0.9396926 , 0.95371693, 0.9659258 ],
        dtype=float32)

• Coordinates: (1)
  • lat
    (lat)
    float32
    75.0 72.5 70.0 ... 20.0 17.5 15.0

    standard_name :

    latitude

    long_name :

    Latitude

    units :

    degrees_north

    axis :

    Y

    array([75. , 72.5, 70. , 67.5, 65. , 62.5, 60. , 57.5, 55. , 52.5, 50. , 47.5,
           45. , 42.5, 40. , 37.5, 35. , 32.5, 30. , 27.5, 25. , 22.5, 20. , 17.5,
           15. ], dtype=float32)

• Indexes: (1)
  • lat
    PandasIndex

    PandasIndex(Index([75.0, 72.5, 70.0, 67.5, 65.0, 62.5, 60.0, 57.5, 55.0, 52.5, 50.0, 47.5,
           45.0, 42.5, 40.0, 37.5, 35.0, 32.5, 30.0, 27.5, 25.0, 22.5, 20.0, 17.5,
           15.0],
          dtype='float32', name='lat'))

• Attributes: (4)

  standard_name :

  latitude

  long_name :

  Latitude

  units :

  degrees_north

  axis :

  Y

Weighted mean

air_weighted = air.weighted(weights)
air_weighted

DataArrayWeighted with weights along dimensions: lat

weighted_mean = air_weighted.mean(("lon", "lat"))
weighted_mean

<xarray.DataArray 'air' (time: 730)>
array([ 6.092401 ,  5.527991 ,  5.6512904,  5.7862387,  5.9117665,
        5.683438 ,  5.9767137,  6.456723 ,  6.5710645,  6.504647 ,
        6.134899 ,  5.926867 ,  5.826823 ,  5.7228684,  5.5780067,
        5.4655237,  5.091241 ,  4.9860153,  5.22863  ,  5.2516623,
        5.4277263,  5.3877935,  5.433898 ,  5.364401 ,  5.4685388,
        5.2290297,  5.350286 ,  5.341831 ,  5.37267  ,  5.3595138,
        5.140337 ,  5.055567 ,  5.0724645,  5.23522  ,  5.3184857,
        5.499176 ,  5.720886 ,  5.7286134,  5.7608094,  5.8255615,
        6.268505 ,  6.436903 ,  6.510232 ,  6.5647664,  6.6087837,
        6.4212675,  5.9147425,  5.554678 ,  5.3292165,  5.3359065,
        5.0705895,  5.283736 ,  5.595219 ,  6.05466  ,  6.5307307,
        6.507418 ,  6.3917427,  6.3951273,  6.398088 ,  6.5293736,
        6.4771113,  6.5357656,  6.692519 ,  6.677368 ,  6.5116334,
        6.447034 ,  6.860378 ,  7.437536 ,  7.6981072,  7.4842625,
        7.2581897,  7.135959 ,  7.0934076,  7.267086 ,  7.348537 ,
        7.321788 ,  7.221145 ,  7.2129273,  7.2840424,  7.5433803,
        7.8543735,  8.11584  ,  8.261897 ,  8.111622 ,  8.219127 ,
        8.358713 ,  8.716147 ,  9.151885 ,  9.370044 ,  9.415864 ,
        9.073437 ,  8.820656 ,  8.804643 ,  8.856381 ,  9.0674515,
        9.407149 ,  9.696928 ,  9.742079 ,  9.659618 ,  9.695612 ,
...
       16.536924 , 16.133308 , 16.05551  , 16.100082 , 15.909406 ,
       15.764092 , 15.631486 , 15.827746 , 16.026222 , 16.31987  ,
       16.156448 , 15.898447 , 15.830862 , 15.810078 , 15.589792 ,
       15.309618 , 15.105176 , 14.964679 , 14.966973 , 14.904601 ,
       14.610661 , 14.330113 , 14.255612 , 14.314032 , 13.940103 ,
       13.758864 , 13.820866 , 14.021832 , 13.888188 , 13.724709 ,
       13.190875 , 12.995149 , 12.66984  , 12.585033 , 12.377668 ,
       12.178653 , 12.082314 , 11.874204 , 11.660165 , 11.601137 ,
       11.558611 , 11.183847 , 11.237345 , 11.091917 , 10.472193 ,
        9.898911 ,  9.431238 ,  9.491593 ,  9.688619 ,  9.998573 ,
        9.793551 ,  9.315285 ,  9.2599325,  9.3849945,  9.343003 ,
        9.202585 ,  9.472327 ,  9.424211 ,  9.050674 ,  8.568184 ,
        7.7191467,  7.3312216,  7.451299 ,  7.4235883,  7.518795 ,
        7.4950356,  7.623864 ,  8.083244 ,  8.04913  ,  8.027269 ,
        8.069612 ,  7.9125304,  8.042945 ,  8.34481  ,  8.507071 ,
        8.708197 ,  8.60495  ,  8.312464 ,  8.257239 ,  7.9841394,
        7.693307 ,  7.421975 ,  7.435237 ,  7.482958 ,  7.6428432,
        7.9084682,  8.036132 ,  7.625418 ,  7.7533154,  7.850425 ,
        7.6213007,  6.8473396,  6.4502645,  5.9852395,  5.5805774],
      dtype=float32)
Coordinates:
  * time     (time) datetime64[ns] 2013-01-01 2013-01-02 ... 2014-12-31

xarray.DataArray

'air'

• time: 730

• 6.092 5.528 5.651 5.786 5.912 5.683 ... 7.621 6.847 6.45 5.985 5.581

  array([ 6.092401 ,  5.527991 ,  5.6512904,  5.7862387,  5.9117665,
          5.683438 ,  5.9767137,  6.456723 ,  6.5710645,  6.504647 ,
          6.134899 ,  5.926867 ,  5.826823 ,  5.7228684,  5.5780067,
          5.4655237,  5.091241 ,  4.9860153,  5.22863  ,  5.2516623,
          5.4277263,  5.3877935,  5.433898 ,  5.364401 ,  5.4685388,
          5.2290297,  5.350286 ,  5.341831 ,  5.37267  ,  5.3595138,
          5.140337 ,  5.055567 ,  5.0724645,  5.23522  ,  5.3184857,
          5.499176 ,  5.720886 ,  5.7286134,  5.7608094,  5.8255615,
          6.268505 ,  6.436903 ,  6.510232 ,  6.5647664,  6.6087837,
          6.4212675,  5.9147425,  5.554678 ,  5.3292165,  5.3359065,
          5.0705895,  5.283736 ,  5.595219 ,  6.05466  ,  6.5307307,
          6.507418 ,  6.3917427,  6.3951273,  6.398088 ,  6.5293736,
          6.4771113,  6.5357656,  6.692519 ,  6.677368 ,  6.5116334,
          6.447034 ,  6.860378 ,  7.437536 ,  7.6981072,  7.4842625,
          7.2581897,  7.135959 ,  7.0934076,  7.267086 ,  7.348537 ,
          7.321788 ,  7.221145 ,  7.2129273,  7.2840424,  7.5433803,
          7.8543735,  8.11584  ,  8.261897 ,  8.111622 ,  8.219127 ,
          8.358713 ,  8.716147 ,  9.151885 ,  9.370044 ,  9.415864 ,
          9.073437 ,  8.820656 ,  8.804643 ,  8.856381 ,  9.0674515,
          9.407149 ,  9.696928 ,  9.742079 ,  9.659618 ,  9.695612 ,
  ...
         16.536924 , 16.133308 , 16.05551  , 16.100082 , 15.909406 ,
         15.764092 , 15.631486 , 15.827746 , 16.026222 , 16.31987  ,
         16.156448 , 15.898447 , 15.830862 , 15.810078 , 15.589792 ,
         15.309618 , 15.105176 , 14.964679 , 14.966973 , 14.904601 ,
         14.610661 , 14.330113 , 14.255612 , 14.314032 , 13.940103 ,
         13.758864 , 13.820866 , 14.021832 , 13.888188 , 13.724709 ,
         13.190875 , 12.995149 , 12.66984  , 12.585033 , 12.377668 ,
         12.178653 , 12.082314 , 11.874204 , 11.660165 , 11.601137 ,
         11.558611 , 11.183847 , 11.237345 , 11.091917 , 10.472193 ,
          9.898911 ,  9.431238 ,  9.491593 ,  9.688619 ,  9.998573 ,
          9.793551 ,  9.315285 ,  9.2599325,  9.3849945,  9.343003 ,
          9.202585 ,  9.472327 ,  9.424211 ,  9.050674 ,  8.568184 ,
          7.7191467,  7.3312216,  7.451299 ,  7.4235883,  7.518795 ,
          7.4950356,  7.623864 ,  8.083244 ,  8.04913  ,  8.027269 ,
          8.069612 ,  7.9125304,  8.042945 ,  8.34481  ,  8.507071 ,
          8.708197 ,  8.60495  ,  8.312464 ,  8.257239 ,  7.9841394,
          7.693307 ,  7.421975 ,  7.435237 ,  7.482958 ,  7.6428432,
          7.9084682,  8.036132 ,  7.625418 ,  7.7533154,  7.850425 ,
          7.6213007,  6.8473396,  6.4502645,  5.9852395,  5.5805774],
        dtype=float32)

• Coordinates: (1)
  • time
    (time)
    datetime64[ns]
    2013-01-01 ... 2014-12-31

    array(['2013-01-01T00:00:00.000000000', '2013-01-02T00:00:00.000000000',
           '2013-01-03T00:00:00.000000000', ..., '2014-12-29T00:00:00.000000000',
           '2014-12-30T00:00:00.000000000', '2014-12-31T00:00:00.000000000'],
          dtype='datetime64[ns]')

• Indexes: (1)
  • time
    PandasIndex

    PandasIndex(DatetimeIndex(['2013-01-01', '2013-01-02', '2013-01-03', '2013-01-04',
                   '2013-01-05', '2013-01-06', '2013-01-07', '2013-01-08',
                   '2013-01-09', '2013-01-10',
                   ...
                   '2014-12-22', '2014-12-23', '2014-12-24', '2014-12-25',
                   '2014-12-26', '2014-12-27', '2014-12-28', '2014-12-29',
                   '2014-12-30', '2014-12-31'],
                  dtype='datetime64[ns]', name='time', length=730, freq='D'))

• Attributes: (0)

Plot: comparison with unweighted mean

Note how the weighted mean temperature is higher than the unweighted.

weighted_mean.plot(label="weighted")
air.mean(("lon", "lat")).plot(label="unweighted")

plt.legend()

<matplotlib.legend.Legend at 0x7f39d61d6680>