Gridding and upward continuation

Most potential field surveys gather data along scattered and uneven flight lines or ground measurements. For a great number of applications we may need to interpolate these data points onto a regular grid at a constant altitude. Upward-continuation is also a routine task for smoothing, noise attenuation, source separation, etc.

Both tasks can be done simultaneously through an equivalent layer [Dampney1969]. We will use harmonica.EQLHarmonic to estimate the coefficients of a set of point sources (the equivalent layer) that fit the observed data. The fitted layer can then be used to predict data values wherever we want, like on a grid at a certain altitude. The sources for EQLHarmonic in particular are placed one beneath each data point at a constant relative depth from the elevation of the data point following [Cooper2000].

The advantage of using an equivalent layer is that it takes into account the 3D nature of the observations, not just their horizontal positions. It also allows data uncertainty to be taken into account and noise to be suppressed though the least-squares fitting process. The main disadvantage is the increased computational load (both in terms of time and memory).

Out:

Number of data points: 7054
Mean height of observations: 541.8293166997448
R² score: 0.9988789670714435

Generated grid:
 <xarray.Dataset>
Dimensions:           (easting: 95, northing: 157)
Coordinates:
  * easting           (easting) float64 -3.24e+05 -3.235e+05 ... -2.769e+05
  * northing          (northing) float64 4.175e+06 4.176e+06 ... 4.253e+06
Data variables:
    magnetic_anomaly  (northing, easting) float64 30.67 30.73 ... 161.3 149.9
Attributes:
    metadata:  Generated by EQLHarmonic(damping=1, points=None, relative_dept...

import matplotlib.pyplot as plt
import numpy as np
import pyproj
import verde as vd
import harmonica as hm


# Fetch the sample total-field magnetic anomaly data from Great Britain
data = hm.datasets.fetch_britain_magnetic()

# Slice a smaller portion of the survey data to speed-up calculations for this
# example
region = [-5.5, -4.7, 57.8, 58.5]
inside = vd.inside((data.longitude, data.latitude), region)
data = data[inside]
print("Number of data points:", data.shape[0])
print("Mean height of observations:", data.altitude_m.mean())

# Since this is a small area, we'll project our data and use Cartesian
# coordinates
projection = pyproj.Proj(proj="merc", lat_ts=data.latitude.mean())
easting, northing = projection(data.longitude.values, data.latitude.values)
coordinates = (easting, northing, data.altitude_m)

# Create the equivalent layer. We'll use the default point source configuration
# at a constant relative depth beneath each observation point. The damping
# parameter helps smooth the predicted data and ensure stability.
eql = hm.EQLHarmonic(relative_depth=1000, damping=1)

# Fit the layer coefficients to the observed magnetic anomaly.
eql.fit(coordinates, data.total_field_anomaly_nt)

# Evaluate the data fit by calculating an R² score against the observed data.
# This is a measure of how well layer the fits the data NOT how good the
# interpolation will be.
print("R² score:", eql.score(coordinates, data.total_field_anomaly_nt))

# Interpolate data on a regular grid with 500 m spacing. The interpolation
# requires an extra coordinate (upward height). By passing in 1500 m, we're
# effectively upward-continuing the data (mean flight height is 500 m).
grid = eql.grid(spacing=500, data_names=["magnetic_anomaly"], extra_coords=1500)

# The grid is a xarray.Dataset with values, coordinates, and metadata
print("\nGenerated grid:\n", grid)

# Plot original magnetic anomaly and the gridded and upward-continued version
fig, (ax1, ax2) = plt.subplots(nrows=1, ncols=2, figsize=(12, 9), sharey=True)

# Get the maximum absolute value between the original and gridded data so we
# can use the same color scale for both plots and have 0 centered at the white
# color.
maxabs = vd.maxabs(data.total_field_anomaly_nt, grid.magnetic_anomaly.values)

ax1.set_title("Observed magnetic anomaly data")
tmp = ax1.scatter(
    easting,
    northing,
    c=data.total_field_anomaly_nt,
    s=20,
    vmin=-maxabs,
    vmax=maxabs,
    cmap="seismic",
)
plt.colorbar(tmp, ax=ax1, label="nT", pad=0.05, aspect=40, orientation="horizontal")
ax1.set_xlim(easting.min(), easting.max())
ax1.set_ylim(northing.min(), northing.max())

ax2.set_title("Gridded and upward-continued")
tmp = grid.magnetic_anomaly.plot.pcolormesh(
    ax=ax2,
    add_colorbar=False,
    add_labels=False,
    vmin=-maxabs,
    vmax=maxabs,
    cmap="seismic",
)
plt.colorbar(tmp, ax=ax2, label="nT", pad=0.05, aspect=40, orientation="horizontal")
ax2.set_xlim(easting.min(), easting.max())
ax2.set_ylim(northing.min(), northing.max())

plt.tight_layout(pad=0)
plt.show()