3D magnetics modelling and inversion

Based on the synthetic model of Li & Oldenburg (1996), we demonstrate 3D inversion of magnetic data. The forward operator bases on the formula given by Holstein et al. (2007).

In the following, we will build the model, create synthetic data, and do inversion using a depth-weighting function as outlined in the paper.

import numpy as np
import pygimli as pg
from pygimli.viewer import pv
from pygimli.physics.gravimetry import MagneticsModelling

Synthetic model and data generation

The grid is 1x1x0.5 km with a spacing of 50 m.

dx = 50
x = np.arange(0., 1001, dx)
y = np.arange(0., 1001, dx)
z = np.arange(0., 501, dx)
grid = pg.createGrid(x, y, z)
print(grid)

Mesh: Nodes: 4851 Cells: 4000 Boundaries: 12800

We create a 3D matrix that is later filled as vector into the grid. The model consists of zeros and patches of 5x6 cells per depth slice that are shifted by one cell for subsequent cells.

v = np.zeros((len(z)-1, len(y)-1, len(x)-1))
for i in range(7):
    v[1+i, 11-i:16-i, 7:13] = 0.05

grid["synth"] = v.ravel()

We show the model making use of the pyvista package that can be called by pygimli.show(). The mesh itself is shown as a wireframe, the anomaly is plotted as a surface plot using a threshold filter. After the first call with hold=True, the plotter is used to draw any subsequent plots that can also be slices or clips. Moreover, the camera position is set so that the vertical axis is going downwards (x is Northing and y is Easting as common in magnetics).

pl, _ = pg.show(grid, style="wireframe", hold=True)
pv.drawMesh(pl, grid, label="synth", style="surface", cMap="Spectral_r",
            filter={"threshold": dict(value=0.05, scalars="synth")})
pl.camera_position = "yz"
pl.camera.roll = 90
pl.camera.azimuth = 180 - 15
pl.camera.elevation = 10
pl.camera.zoom(1.2)
_ = pl.show()

For the computation of the total field, we define the global magnetic field using the IGRF (total field, inclination and declination) settings given in the paper. Any global field can also be retrieved by the pyIGRF module.

F, I, D = 50000e-9, 75, 25
H = F * np.cos(np.deg2rad(I))
X = H * np.cos(np.deg2rad(D))
Y = H * np.sin(np.deg2rad(D))
Z = F * np.sin(np.deg2rad(I))
igrf = [D, I, H, X, Y, Z, F]

py, px = np.meshgrid(x, y)
px = px.ravel()
py = py.ravel()
points = np.column_stack((px, py, -np.ones_like(px)*20))

# The forward operator
cmp = ["TFA"]  # ["Bx", "By", "Bz"]
fop = MagneticsModelling(grid, points, cmp, igrf)
model = pg.Vector(grid.cellCount(), 1.0)
data = fop.response(grid["synth"])

[++++++++++++++++ 100% ++++++++++++++++] 441 of 441 complete

Just like in the paper, the data are contaminated with an error model consisting of relative and absolute noise of 2% and 1 nT, respectively.

err = 0.01
noise_level = 1e-9
data += np.random.randn(len(data)) * noise_level

Depth weighting

In the paper of Li & Oldenburg (1996), they propose a depth weighting of the constraints with the formula

\[w_z = \frac{1}{(z+z_0)^{3/2}}\]

# depth weighting
bz = np.array([b.center().z() for b in grid.boundaries() if not b.outside()])
z0 = 25
wz = 10 / (bz+z0)**1.5

Inversion

The inversion is rather straightforward using the standard inversion framework pygimli.Inversion.

inv = pg.Inversion(fop=fop, verbose=True)  # , debug=True)
inv.setRegularization(limits=[0, 0.07])  # to limit values
inv.setConstraintWeights(wz)
invmodel = inv.run(data, absoluteError=noise_level, lam=1e4, startModel=1e-3, verbose=True)
grid["inv"] = invmodel

fop: <pygimli.physics.gravimetry.MagneticsModelling.MagneticsModelling object at 0x7ff374fc6630>
Data transformation: <pygimli.core._pygimli_.RTrans object at 0x7ff3d58108e0>
Model transformation (cumulative):
         0 <pygimli.core._pygimli_.RTransLogLU object at 0x7ff3d58108e0>
min/max (data): -7.8e-08/5.6e-07
min/max (error): 1.18%/811%
min/max (start model): 1.0e-03/1.0e-03
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
inv.iter 1 ... chi² = 1614.53 (dPhi = -37.53%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 2 ... chi² = 314.86 (dPhi = 80.37%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 3 ... chi² = 30.79 (dPhi = 89.55%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 4 ... chi² = 11.25 (dPhi = 59.56%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 5 ... chi² = 1.51 (dPhi = 71.56%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 6 ... chi² = 1.15 (dPhi = 11.57%) lam: 10000.0
--------------------------------------------------------------------------------
inv.iter 7 ... chi² = 0.71 (dPhi = 12.17%) lam: 10000.0


################################################################################
#                  Abort criterion reached: chi² <= 1 (0.71)                   #
################################################################################

Visualization

For showing the model, we again use the threshold filter with a value of 0.02. For comparison with the synthetic model, we plot the latter as a wireframe.

pl, _ = pg.show(grid, label="synth", style="wireframe", hold=True,
                filter={"threshold": dict(value=0.025, scalars="synth")})
pv.drawMesh(pl, grid, label="inv", style="surface", cMap="Spectral_r",
            filter={"threshold": dict(value=0.02, scalars="inv")})
pv.drawMesh(pl, grid, label="inv", style="surface", cMap="Spectral_r",
            filter={"slice": dict(normal=[-1, 0, 0], origin=[500, 600, 250])})
pl.camera_position = "yz"
pl.camera.roll = 90
pl.camera.azimuth = 180 - 15
pl.camera.elevation = 10
pl.camera.zoom(1.2)
_ = pl.show()

The model can nicely outline the top part of the anomalous body, but not its depth extent.

We compare the data and model response by means of scatter plots:

fig, ax = pg.plt.subplots(ncols=2, figsize=(12, 5), sharex=True, sharey=True)
vals = data * 1e9
mm = np.max(np.abs(vals))
ax[0].scatter(px, py, c=vals, cmap="bwr", vmin=-mm, vmax=mm);
_ = ax[1].scatter(px, py, c=inv.response*1e9, cmap="bwr", vmin=-mm, vmax=mm);

Alternatively, we can also plot the error-weighted misfit.

misfit = (inv.response - data) / noise_level
_ = pg.plt.scatter(py, px, c=misfit, cmap="bwr", vmin=-3, vmax=3);

References

• Li, Y. & Oldenburg, D. (1996): 3-D inversion of magnetic data. Geophysics 61(2), 394-408.

• Holstein, H., Sherratt, E.M., Reid, A.B.  (2007): Gravimagnetic field tensor gradiometry formulas for uniform polyhedra, SEG Ext. Abstr.