Assessing levelling results

8. Assessing levelling results#

[1]:

%load_ext autoreload
%autoreload 2


import logging

import geopandas as gpd
import harmonica as hm
import numpy as np
import pandas as pd
import plotly.io as pio
import verde as vd

import airbornegeo

logging.getLogger("airbornegeo").setLevel("INFO")
logging.basicConfig()
pio.renderers.default = "notebook"

/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:326: SyntaxWarning: invalid escape sequence '\m'
  daft = \mathbb{F}(da - \overline{da})
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:497: SyntaxWarning: invalid escape sequence '\m'
  da = \mathbb{F}(daft - \overline{daft})
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:692: SyntaxWarning: invalid escape sequence '\o'
  da' = da - \overline{da}
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:767: SyntaxWarning: invalid escape sequence '\o'
  da1' = da1 - \overline{da1};\ \ da2' = da2 - \overline{da2}
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:845: SyntaxWarning: invalid escape sequence '\o'
  da1' = da1 - \overline{da1};\ \ da2' = da2 - \overline{da2}
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:954: SyntaxWarning: invalid escape sequence '\s'
  \text{iso}_{ps} = k_r N^{-1} \sum_{N} |\mathbb{F}(da')|^2
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:1032: SyntaxWarning: invalid escape sequence '\s'
  \text{iso}_{ps} = k_r N^{-1} \sum_{N} |\mathbb{F}(da')|^2
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:1118: SyntaxWarning: invalid escape sequence '\s'
  \text{iso}_{cs} = k_r N^{-1} \sum_{N} (\mathbb{F}(da1') {\mathbb{F}(da2')}^*)

8.1. Load data#

This is a subset of the BAS AGAP survey over Antarctica’s Gamburtsev Subglacial Mountains. The file is download and subset in the notebook AGAP_magnetic_survey, and the BAS processing steps are repeated in the notebook processing_AGAP_magnetic_survey.

[2]:

data_df = pd.read_csv("data/AGAP_magnetic_survey_processed_blocked.csv")
data_df = data_df[
    [
        "easting",
        "northing",
        "height",
        "line",
        "unixtime",
        "distance_along_line",
        "mag",
    ]
]

# for testing limit number of lines
data_df = data_df[~data_df.line.between(133, 142)]
data_df = data_df[~data_df.line.between(168, 176)]
data_df = data_df[
    (data_df.line.isin(data_df.line.unique()[::2])) | (data_df.line >= 143)
]

# define flight lines vs tie lines with column 'tie' which is True or False
data_df["tie"] = False
data_df.loc[data_df.line >= 142, "tie"] = True

data_df

[2]:
easting northing height line unixtime distance_along_line mag tie
0 6.210991e+05 159056.748193 4110.45 1 1.229500e+09 27.181565 -33.385 False
1 6.212065e+05 159071.301512 4114.50 1 1.229500e+09 135.587530 -35.120 False
2 6.213138e+05 159085.113599 4117.90 1 1.229500e+09 243.749367 -36.875 False
3 6.214207e+05 159099.184618 4120.80 1 1.229500e+09 351.600337 -38.585 False
4 6.215272e+05 159114.303863 4123.15 1 1.229500e+09 459.104857 -40.205 False
... ... ... ... ... ... ... ... ...
703796 1.082115e+06 112573.528454 4191.50 206 1.231249e+09 177936.928871 -27.040 True
703797 1.082103e+06 112648.360155 4191.60 206 1.231249e+09 178012.800558 -28.040 True
703798 1.082084e+06 112760.548617 4191.80 206 1.231249e+09 178126.596403 -29.410 True
703799 1.082065e+06 112872.636385 4192.30 206 1.231249e+09 178240.329983 -30.840 True
703800 1.082042e+06 113004.019427 4193.40 206 1.231249e+09 178373.722649 -40.840 True

399208 rows × 8 columns

8.2. Find intersections#

[3]:

# convert dataframe into geodataframe
data_df = gpd.GeoDataFrame(
    data_df,
    geometry=gpd.points_from_xy(data_df.easting, data_df.northing),
    crs="EPSG:3031",
)

[4]:

# calculate theoretical intersection points
inters = airbornegeo.create_intersection_table(
    data_df,
    plot_map=False,
    plot_hist=False,
)
inters

INFO:airbornegeo:found 662 intersections

[4]:
line tie geometry max_dist easting northing
0 1 143 POINT (1158153 254083) 16.441255 1158153.0 254083.0
1 1 144 POINT (1190820 259865) 24.124887 1190820.0 259865.0
2 1 145 POINT (1223524 265689) 17.269650 1223524.0 265689.0
3 1 146 POINT (1256193 271497) 31.608179 1256193.0 271497.0
4 1 147 POINT (1288901 277249) 50.174377 1288901.0 277249.0
... ... ... ... ... ... ...
657 125 199 POINT (1318165 302398) 33.689493 1318165.0 302398.0
658 125 200 POINT (1350929 308239) 36.343660 1350929.0 308239.0
659 127 148 POINT (1319894 292699) 22.093103 1319894.0 292699.0
660 127 199 POINT (1319922 292702) 23.297236 1319922.0 292702.0
661 127 200 POINT (1352642 298507) 36.743873 1352642.0 298507.0

662 rows × 6 columns

8.3. Add intersections as rows to the dataframe#

[5]:

data_df, inters = airbornegeo.interpolate_intersections(
    data_df,
    inters,
    to_interp=["mag", "height"],
    window_width=500,
    method="cubic",
    extrapolate=False,
)

[6]:

lines_without_inters = airbornegeo.lines_without_intersections(data_df, inters)
lines_without_inters

[6]:

[np.int64(107),
 np.int64(109),
 np.int64(129),
 np.int64(131),
 np.int64(188),
 np.int64(189),
 np.int64(190),
 np.int64(192),
 np.int64(193),
 np.int64(194),
 np.int64(203)]

[7]:

# drop lines without intersections
data_df = data_df[~data_df.line.isin(lines_without_inters)]

8.4. Calculate initial cross-over errors#

[8]:

inters = airbornegeo.calculate_crossover_errors(
    data_df,
    inters,
    data_col="mag",
    plot_map=True,
)
_images/assessing_levelling_results_12_2.png 
[9]:

inters.head()

[9]:
line tie geometry max_dist easting northing dist_along_flight_line dist_along_flight_tie flight_interpolation_type tie_interpolation_type flight_height tie_height mistie_0
0 1 143 POINT (1158153 254083) 16.441255 1158153.0 254083.0 545768.072695 138071.419338 interpolated interpolated 4168.038983 4008.397691 100.458661
1 1 144 POINT (1190820 259865) 24.124887 1190820.0 259865.0 578952.394781 131585.125284 interpolated interpolated 4174.500813 4010.840823 32.151483
2 1 145 POINT (1223524 265689) 17.269650 1223524.0 265689.0 612181.972720 147253.496664 interpolated interpolated 3544.149925 3995.961216 72.936553
3 1 146 POINT (1256193 271497) 31.608179 1256193.0 271497.0 645372.035087 136211.666619 interpolated interpolated 3564.579994 3578.174277 84.125118
4 1 147 POINT (1288901 277249) 50.174377 1288901.0 277249.0 678598.170740 155115.224990 interpolated interpolated 3536.160719 3545.947314 -7.731396

8.5. Grid the pre-levelled data#

[10]:

# block reduce the line data
data_df_blocked = airbornegeo.block_reduce(
    data_df,
    np.median,
    spacing=2000,
    reduce_by="distance_along_line",
    groupby_column="line",
)
data_df_blocked = data_df_blocked.dropna(
    subset=["easting", "northing", "height", "mag"]
)
data_df_blocked.describe()

[10]:
distance_along_line easting northing height unixtime mag intersecting_line line
count 18198.000000 1.819800e+04 18198.000000 18198.000000 1.698300e+04 18198.000000 0.0 18198.000000
mean 190481.664973 1.010550e+06 235793.150394 3869.887801 1.230898e+09 -12.042309 NaN 98.511155
std 136436.359312 2.580533e+05 146263.995765 438.825066 6.573929e+05 88.895060 NaN 62.363794
min 932.773718 5.328906e+05 -302813.880978 2478.775000 1.229500e+09 -362.240000 NaN 1.000000
25% 83307.405473 7.839629e+05 150247.851427 3648.200000 1.230456e+09 -65.695000 NaN 49.000000
50% 167343.130074 9.961314e+05 233907.068600 4008.825000 1.230997e+09 -22.415000 NaN 85.000000
75% 263162.179086 1.231160e+06 324823.087590 4201.243750 1.231484e+09 27.578125 NaN 156.000000
max 744852.096684 1.608560e+06 695692.176078 4495.200000 1.232150e+09 833.162500 NaN 206.000000 
[11]:

coords = (
    data_df_blocked.easting,
    data_df_blocked.northing,
    data_df_blocked.height,
)
eqs_unlevelled = hm.EquivalentSources(damping=0.1, depth="default", block_size=2000)
eqs_unlevelled.fit(coords, data_df_blocked.mag)

[11]:

EquivalentSources(block_size=2000, damping=0.1)
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[12]:

# Define grid coordinates
grid_coords = vd.grid_coordinates(
    region=vd.pad_region(vd.get_region(coords), 10e3),
    spacing=2000,
    extra_coords=4200,  # upward continue
    adjust="region",
)
grid_unlevelled = eqs_unlevelled.grid(grid_coords)

grid_unlevelled = vd.distance_mask(
    (coords[0], coords[1]), maxdist=10e3, grid=grid_unlevelled
)
grid_unlevelled = grid_unlevelled.reset_coords(names="upward").scalars
grid_unlevelled.plot(robust=True)

[12]:

<matplotlib.collections.QuadMesh at 0x7f0753089d30>
_images/assessing_levelling_results_17_1.png

8.6. Level the survey#

[13]:

data_df, inters = airbornegeo.alternating_iterative_line_levelling(
    data_df,
    inters,
    data_col="mag",
    levelled_col="mag",
    degree=0,
    iterations=5,
)
inters.head()

[13]:
line tie geometry max_dist easting northing dist_along_flight_line dist_along_flight_tie flight_interpolation_type tie_interpolation_type ... mistie_1 mistie_2 mistie_3 mistie_4 mistie_5 mistie_6 mistie_7 mistie_8 mistie_9 mistie_10
0 1 143 POINT (1158153 254083) 16.441255 1158153.0 254083.0 545768.072695 138071.419338 interpolated interpolated ... 96.945621 72.958517 71.904752 67.038524 67.150418 63.393096 63.723800 60.721570 61.069625 58.628182
1 1 144 POINT (1190820 259865) 24.124887 1190820.0 259865.0 578952.394781 131585.125284 interpolated interpolated ... 28.638444 19.088634 18.034870 13.168642 13.280535 9.523214 9.853917 6.851688 7.199742 4.758299
2 1 145 POINT (1223524 265689) 17.269650 1223524.0 265689.0 612181.972720 147253.496664 interpolated interpolated ... 69.423514 49.368210 48.314446 43.448218 43.560111 39.802790 40.133493 37.131264 37.479318 35.037876
3 1 146 POINT (1256193 271497) 31.608179 1256193.0 271497.0 645372.035087 136211.666619 interpolated interpolated ... 80.612078 60.446633 59.392868 54.526640 54.638534 50.881212 51.211916 48.209686 48.557741 46.116298
4 1 147 POINT (1288901 277249) 50.174377 1288901.0 277249.0 678598.170740 155115.224990 interpolated interpolated ... -11.244436 -2.791611 -3.845375 -8.711603 -8.599710 -12.357032 -12.026328 -15.028558 -14.680503 -17.121946

5 rows × 23 columns

[14]:

airbornegeo.plot_levelling_convergence(inters)
_images/assessing_levelling_results_20_1.png 
[15]:

inters = airbornegeo.calculate_crossover_errors(
    data_df,
    inters,
    data_col="mag",
    plot_map=True,
)
_images/assessing_levelling_results_21_1.png 
[16]:

# block reduce the line data
data_df_blocked = airbornegeo.block_reduce(
    data_df,
    np.median,
    spacing=2000,
    reduce_by="distance_along_line",
    groupby_column="line",
)
data_df_blocked = data_df_blocked.dropna(
    subset=["easting", "northing", "height", "mag"]
)
data_df_blocked.describe()

[16]:
distance_along_line easting northing height unixtime mag intersecting_line line
count 18198.000000 1.819800e+04 18198.000000 18198.000000 1.698300e+04 18198.000000 0.0 18198.000000
mean 190481.664973 1.010550e+06 235793.150394 3869.887801 1.230898e+09 -18.408009 NaN 98.511155
std 136436.359312 2.580533e+05 146263.995765 438.825066 6.573929e+05 85.083158 NaN 62.363794
min 932.773718 5.328906e+05 -302813.880978 2478.775000 1.229500e+09 -338.586624 NaN 1.000000
25% 83307.405473 7.839629e+05 150247.851427 3648.200000 1.230456e+09 -69.924084 NaN 49.000000
50% 167343.130074 9.961314e+05 233907.068600 4008.825000 1.230997e+09 -30.884226 NaN 85.000000
75% 263162.179086 1.231160e+06 324823.087590 4201.243750 1.231484e+09 17.898138 NaN 156.000000
max 744852.096684 1.608560e+06 695692.176078 4495.200000 1.232150e+09 794.541776 NaN 206.000000 
[17]:

coords = (
    data_df_blocked.easting,
    data_df_blocked.northing,
    data_df_blocked.height,
)
eqs_levelled_trend_0 = hm.EquivalentSources(
    damping=0.1, depth="default", block_size=2000
)
eqs_levelled_trend_0.fit(coords, data_df_blocked.mag)

[17]:

EquivalentSources(block_size=2000, damping=0.1)
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[18]:

grid_levelled_trend_0 = eqs_levelled_trend_0.grid(grid_coords)

grid_levelled_trend_0 = vd.distance_mask(
    (coords[0], coords[1]), maxdist=10e3, grid=grid_levelled_trend_0
)
grid_levelled_trend_0 = grid_levelled_trend_0.reset_coords(names="upward").scalars
grid_levelled_trend_0.plot(robust=True)

[18]:

<matplotlib.collections.QuadMesh at 0x7f07002f9810>
_images/assessing_levelling_results_24_1.png 
[19]:

(grid_unlevelled - grid_levelled_trend_0).plot(robust=True)

[19]:

<matplotlib.collections.QuadMesh at 0x7f07003625d0>
_images/assessing_levelling_results_25_1.png

8.7. Repeat with a trend order of 1#

[20]:

data_df, inters = airbornegeo.alternating_iterative_line_levelling(
    data_df,
    inters,
    data_col="mag",
    levelled_col="mag",
    degree=1,
    iterations=5,
)
inters.head()

[20]:
line tie geometry max_dist easting northing dist_along_flight_line dist_along_flight_tie flight_interpolation_type tie_interpolation_type ... mistie_11 mistie_12 mistie_13 mistie_14 mistie_15 mistie_16 mistie_17 mistie_18 mistie_19 mistie_20
0 1 143 POINT (1158153 254083) 16.441255 1158153.0 254083.0 545768.072695 138071.419338 interpolated interpolated ... 55.628017 50.833479 48.668087 46.896275 45.560415 44.541105 43.723455 43.023456 42.508024 41.987979
1 1 144 POINT (1190820 259865) 24.124887 1190820.0 259865.0 578952.394781 131585.125284 interpolated interpolated ... 1.119289 -0.019764 -2.416675 -3.516709 -4.981390 -5.637918 -6.532933 -6.976960 -7.541316 -7.855782
2 1 145 POINT (1223524 265689) 17.269650 1223524.0 265689.0 612181.972720 147253.496664 interpolated interpolated ... 30.759149 30.141541 27.512797 27.085037 25.491359 25.197815 24.225328 24.037369 23.424022 23.315179
3 1 146 POINT (1256193 271497) 31.608179 1256193.0 271497.0 645372.035087 136211.666619 interpolated interpolated ... 41.198616 39.414904 36.554601 36.801009 35.078487 35.149071 34.099204 34.168111 33.505831 33.603219
4 1 147 POINT (1288901 277249) 50.174377 1288901.0 277249.0 678598.170740 155115.224990 interpolated interpolated ... -22.679278 -21.316610 -24.408724 -23.490614 -25.342120 -24.908710 -26.036040 -25.711101 -26.422367 -26.119346

5 rows × 33 columns

[21]:

airbornegeo.plot_levelling_convergence(inters)
_images/assessing_levelling_results_28_1.png 
[22]:

inters = airbornegeo.calculate_crossover_errors(
    data_df,
    inters,
    data_col="mag",
    plot_map=True,
)
_images/assessing_levelling_results_29_1.png 
[23]:

# block reduce the line data
data_df_blocked = airbornegeo.block_reduce(
    data_df,
    np.median,
    spacing=2000,
    reduce_by="distance_along_line",
    groupby_column="line",
)
data_df_blocked = data_df_blocked.dropna(
    subset=["easting", "northing", "height", "mag"]
)
data_df_blocked.describe()

[23]:
distance_along_line easting northing height unixtime mag intersecting_line line
count 18198.000000 1.819800e+04 18198.000000 18198.000000 1.698300e+04 18198.000000 0.0 18198.000000
mean 190481.664973 1.010550e+06 235793.150394 3869.887801 1.230898e+09 -19.017283 NaN 98.511155
std 136436.359312 2.580533e+05 146263.995765 438.825066 6.573929e+05 99.987430 NaN 62.363794
min 932.773718 5.328906e+05 -302813.880978 2478.775000 1.229500e+09 -684.449432 NaN 1.000000
25% 83307.405473 7.839629e+05 150247.851427 3648.200000 1.230456e+09 -70.099556 NaN 49.000000
50% 167343.130074 9.961314e+05 233907.068600 4008.825000 1.230997e+09 -29.697816 NaN 85.000000
75% 263162.179086 1.231160e+06 324823.087590 4201.243750 1.231484e+09 21.574995 NaN 156.000000
max 744852.096684 1.608560e+06 695692.176078 4495.200000 1.232150e+09 805.336305 NaN 206.000000 
[24]:

coords = (
    data_df_blocked.easting,
    data_df_blocked.northing,
    data_df_blocked.height,
)
eqs_levelled_trend_1 = hm.EquivalentSources(
    damping=0.1, depth="default", block_size=2000
)
eqs_levelled_trend_1.fit(coords, data_df_blocked.mag)

[24]:

EquivalentSources(block_size=2000, damping=0.1)
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[25]:

grid_levelled_trend_1 = eqs_levelled_trend_1.grid(grid_coords)

grid_levelled_trend_1 = vd.distance_mask(
    (coords[0], coords[1]), maxdist=10e3, grid=grid_levelled_trend_1
)
grid_levelled_trend_1 = grid_levelled_trend_1.reset_coords(names="upward").scalars
grid_levelled_trend_1.plot(robust=True)

[25]:

<matplotlib.collections.QuadMesh at 0x7f070020a350>
_images/assessing_levelling_results_32_1.png 
[26]:

(grid_unlevelled - grid_levelled_trend_1).plot(robust=True)

[26]:

<matplotlib.collections.QuadMesh at 0x7f0700156fd0>
_images/assessing_levelling_results_33_1.png 
[27]:

(grid_levelled_trend_0 - grid_levelled_trend_1).plot(robust=True)

[27]:

<matplotlib.collections.QuadMesh at 0x7f0700293d90>
_images/assessing_levelling_results_34_1.png

8.8. Use spatial derivatives to assess levelling performance#

[29]:

import invert4geom
import polartoolkit as ptk

titles = [
    "Unlevelled ",
    "Levelled trend 0",
    "Levelled trend 1",
]
for i, g in enumerate(
    [
        grid_unlevelled,
        grid_levelled_trend_0,
        grid_levelled_trend_1,
    ]
):
    east_deriv = invert4geom.filter_grid(
        g,
        filter_type="easting_deriv",
    )
    north_deriv = invert4geom.filter_grid(g, filter_type="northing_deriv")
    up_deriv = invert4geom.filter_grid(
        g,
        filter_type="up_deriv",
    )
    total_gradient = invert4geom.filter_grid(
        g,
        filter_type="total_gradient",
    )

    fig = ptk.subplots(
        [east_deriv, north_deriv, up_deriv, total_gradient],
        fig_title=titles[i],
        titles=[
            "Easting derivative",
            "Northing derivative",
            "Upward derivative",
            "Total gradient",
        ],
        cmap="balance+h0",
        robust=True,
        fig_height=10,
    )
    fig.show()

/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
_images/assessing_levelling_results_36_1.png 
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
_images/assessing_levelling_results_36_3.png 
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/harmonica/filters/_fft.py:48: FutureWarning: dropping variables using `drop` is deprecated; use drop_vars.
  grid = grid.drop(bad_coords)
/home/airbornegeo/airbornegeo/.pixi/envs/default/lib/python3.13/site-packages/xrft/xrft.py:560: FutureWarning: Default ifft's behaviour (lag=None) changed! Default value of lag was zero (centered output coordinates) and is now set to transformed coordinate's attribute: 'direct_lag'.
  warnings.warn(msg, FutureWarning)
_images/assessing_levelling_results_36_5.png

The above plots show the easting, northing, and upward derivatives, and the total gradients, for the unlevelled grid, and the grid levelled with trend orders 0 and 1. These highlight the levelling errors in the data. These errors appear to be significantly reduced with the levelling.