Maps: ComboMaps#

Invert synthetic magnetic data with variable background values and a single block anomaly buried at depth. We will use the Sum Map to invert for both the background values and an heterogeneous susceptibiilty model.

1
  • plot sumMap
  • plot sumMap
Running inversion with SimPEG v0.24.1.dev21+gd58b57fa8

                    simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                    ***Done using the default solver Pardiso and no solver_opts.***

model has any nan: 0
=============================== Projected GNCG ===============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
   0  6.06e+05  9.12e+06  4.27e-04  9.12e+06    6.29e+01      0
   1  3.03e+05  1.33e+05  7.22e-02  1.55e+05    3.47e+01      0
   2  1.51e+05  4.94e+04  2.22e-01  8.30e+04    4.84e+01      0
   3  7.57e+04  2.13e+04  3.51e-01  4.79e+04    5.44e+01      0   Skip BFGS
   4  3.79e+04  7.06e+03  4.67e-01  2.47e+04    4.87e+01      0   Skip BFGS
   5  1.89e+04  4.14e+03  5.43e-01  1.44e+04    3.64e+01      0   Skip BFGS
   6  9.47e+03  1.04e+03  5.98e-01  6.70e+03    4.81e+01      0
   7  4.73e+03  1.04e+03  5.98e-01  3.87e+03    4.98e+01      6   Skip BFGS
   8  2.37e+03  4.82e+02  6.46e-01  2.01e+03    4.44e+01      0
   9  1.18e+03  4.72e+02  6.49e-01  1.24e+03    3.49e+01      2   Skip BFGS
  10  5.92e+02  4.17e+02  6.69e-01  8.13e+02    4.75e+01      0
  11  2.96e+02  4.17e+02  6.69e-01  6.15e+02    4.92e+01     11   Skip BFGS
  12  1.48e+02  4.14e+02  6.73e-01  5.14e+02    3.71e+01      2
  13  7.40e+01  4.03e+02  6.94e-01  4.55e+02    3.91e+01      0   Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 0.010193003081995776
irls_threshold 0.012255346444106951
  14  7.40e+01  3.98e+02  9.59e-01  4.69e+02    5.62e+01      3
  15  7.40e+01  3.98e+02  1.03e+00  4.74e+02    5.85e+01      4   Skip BFGS
  16  7.40e+01  3.97e+02  1.08e+00  4.77e+02    4.43e+01      4   Skip BFGS
  17  7.40e+01  3.97e+02  1.11e+00  4.80e+02    3.63e+01      4   Skip BFGS
  18  7.40e+01  3.97e+02  1.14e+00  4.81e+02    6.08e+01      0
  19  7.40e+01  3.97e+02  1.14e+00  4.81e+02    6.08e+01     11   Skip BFGS
  20  7.40e+01  3.95e+02  1.12e+00  4.78e+02    3.50e+01      2
  21  7.40e+01  3.93e+02  1.09e+00  4.74e+02    3.86e+01      0   Skip BFGS
  22  7.40e+01  3.89e+02  1.05e+00  4.67e+02    5.58e+01      3
  23  7.40e+01  3.89e+02  9.92e-01  4.63e+02    5.59e+01     11   Skip BFGS
  24  7.40e+01  3.89e+02  9.27e-01  4.58e+02    3.59e+01      4
  25  7.40e+01  3.90e+02  8.52e-01  4.53e+02    4.07e+01      2   Skip BFGS
  26  7.40e+01  3.90e+02  7.78e-01  4.48e+02    6.00e+01      3
  27  7.40e+01  3.90e+02  7.04e-01  4.42e+02    6.12e+01      3   Skip BFGS
  28  7.40e+01  3.89e+02  6.32e-01  4.36e+02    3.82e+01      3   Skip BFGS
  29  7.40e+01  3.90e+02  5.63e-01  4.31e+02    3.64e+01      3   Skip BFGS
  30  7.40e+01  3.86e+02  4.91e-01  4.22e+02    6.15e+01      1
  31  7.40e+01  3.86e+02  4.33e-01  4.18e+02    6.17e+01      5   Skip BFGS
  32  7.40e+01  3.86e+02  3.77e-01  4.13e+02    3.68e+01      2
  33  7.40e+01  3.82e+02  3.16e-01  4.05e+02    6.30e+01      0
Reach maximum number of IRLS cycles: 20
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 9.1187e+05
1 : |xc-x_last| = 1.8099e-03 <= tolX*(1+|x0|) = 1.0075e-01
0 : |proj(x-g)-x|    = 6.3001e+01 <= tolG          = 1.0000e-03
0 : |proj(x-g)-x|    = 6.3001e+01 <= 1e3*eps       = 1.0000e-03
0 : maxIter   =     100    <= iter          =     34
------------------------- DONE! -------------------------

from discretize import TensorMesh
from discretize.utils import active_from_xyz
from simpeg import (
    utils,
    maps,
    regularization,
    data_misfit,
    optimization,
    inverse_problem,
    directives,
    inversion,
)
from simpeg.potential_fields import magnetics
import numpy as np
import matplotlib.pyplot as plt


def run(plotIt=True):
    h0_amplitude, h0_inclination, h0_declination = (50000.0, 90.0, 0.0)

    # Create a mesh
    dx = 5.0

    hxind = [(dx, 5, -1.3), (dx, 10), (dx, 5, 1.3)]
    hyind = [(dx, 5, -1.3), (dx, 10), (dx, 5, 1.3)]
    hzind = [(dx, 5, -1.3), (dx, 10)]

    mesh = TensorMesh([hxind, hyind, hzind], "CCC")

    # Lets create a simple Gaussian topo and set the active cells
    [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y)
    zz = -np.exp((xx**2 + yy**2) / 75**2) + mesh.nodes_z[-1]

    # We would usually load a topofile
    topo = np.c_[utils.mkvc(xx), utils.mkvc(yy), utils.mkvc(zz)]

    # Go from topo to array of indices of active cells
    actv = active_from_xyz(mesh, topo, "N")
    nC = int(actv.sum())
    # Create and array of observation points
    xr = np.linspace(-20.0, 20.0, 20)
    yr = np.linspace(-20.0, 20.0, 20)
    X, Y = np.meshgrid(xr, yr)

    # Move the observation points 5m above the topo
    Z = -np.exp((X**2 + Y**2) / 75**2) + mesh.nodes_z[-1] + 5.0

    # Create a MAGsurvey
    rxLoc = np.c_[utils.mkvc(X.T), utils.mkvc(Y.T), utils.mkvc(Z.T)]
    rxLoc = magnetics.Point(rxLoc)
    srcField = magnetics.UniformBackgroundField(
        receiver_list=[rxLoc],
        amplitude=h0_amplitude,
        inclination=h0_inclination,
        declination=h0_declination,
    )
    survey = magnetics.Survey(srcField)

    # We can now create a susceptibility model and generate data
    model = np.zeros(mesh.nC)

    # Change values in half the domain
    model[mesh.gridCC[:, 0] < 0] = 0.01

    # Add a block in half-space
    model = utils.model_builder.add_block(
        mesh.gridCC, model, np.r_[-10, -10, 20], np.r_[10, 10, 40], 0.05
    )

    model = utils.mkvc(model)
    model = model[actv]

    # Create active map to go from reduce set to full
    actvMap = maps.InjectActiveCells(mesh, actv, np.nan)

    # Create reduced identity map
    idenMap = maps.IdentityMap(nP=nC)

    # Create the forward model operator
    prob = magnetics.Simulation3DIntegral(
        mesh,
        survey=survey,
        chiMap=idenMap,
        active_cells=actv,
        store_sensitivities="forward_only",
    )

    # Compute linear forward operator and compute some data
    data = prob.make_synthetic_data(
        model, relative_error=0.0, noise_floor=1, add_noise=True
    )

    # Create a homogenous maps for the two domains
    domains = [mesh.gridCC[actv, 0] < 0, mesh.gridCC[actv, 0] >= 0]
    homogMap = maps.SurjectUnits(domains)

    # Create a wire map for a second model space, voxel based
    wires = maps.Wires(("homo", len(domains)), ("hetero", nC))

    # Create Sum map
    sumMap = maps.SumMap([homogMap * wires.homo, wires.hetero])

    # Create the forward model operator
    prob = magnetics.Simulation3DIntegral(
        mesh, survey=survey, chiMap=sumMap, active_cells=actv, store_sensitivities="ram"
    )

    # Make sensitivity weighting
    # Take the cell number out of the scaling.
    # Want to keep high sens for large volumes
    wr = (
        prob.getJtJdiag(np.ones(sumMap.shape[1]))
        / np.r_[homogMap.P.T * mesh.cell_volumes[actv], mesh.cell_volumes[actv]] ** 2.0
    )
    # Scale the model spaces independently
    wr[wires.homo.index] /= np.max((wires.homo * wr)) * utils.mkvc(
        homogMap.P.sum(axis=0).flatten()
    )
    wr[wires.hetero.index] /= np.max(wires.hetero * wr)
    wr = wr**0.5

    ## Create a regularization
    # For the homogeneous model
    regMesh = TensorMesh([len(domains)])

    reg_m1 = regularization.Sparse(regMesh, mapping=wires.homo)
    reg_m1.set_weights(weights=wires.homo * wr)

    reg_m1.norms = [0, 2]
    reg_m1.reference_model = np.zeros(sumMap.shape[1])

    # Regularization for the voxel model
    reg_m2 = regularization.Sparse(
        mesh, active_cells=actv, mapping=wires.hetero, gradient_type="components"
    )
    reg_m2.set_weights(weights=wires.hetero * wr)

    reg_m2.norms = [0, 0, 0, 0]
    reg_m2.reference_model = np.zeros(sumMap.shape[1])

    reg = reg_m1 + reg_m2

    # Data misfit function
    dmis = data_misfit.L2DataMisfit(simulation=prob, data=data)

    # Add directives to the inversion
    opt = optimization.ProjectedGNCG(
        maxIter=100,
        lower=0.0,
        upper=1.0,
        maxIterLS=20,
        maxIterCG=10,
        tolCG=1e-3,
        tolG=1e-3,
        eps=1e-6,
    )
    invProb = inverse_problem.BaseInvProblem(dmis, reg, opt)
    betaest = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)

    # Here is where the norms are applied
    # Use pick a threshold parameter empirically based on the distribution of
    #  model parameters
    IRLS = directives.UpdateIRLS(f_min_change=1e-3)

    update_Jacobi = directives.UpdatePreconditioner()
    inv = inversion.BaseInversion(invProb, directiveList=[IRLS, betaest, update_Jacobi])

    # Run the inversion
    m0 = np.ones(sumMap.shape[1]) * 1e-4  # Starting model
    prob.model = m0
    mrecSum = inv.run(m0)
    if plotIt:
        mesh.plot_3d_slicer(
            actvMap * model,
            aspect="equal",
            zslice=30,
            pcolor_opts={"cmap": "inferno_r"},
            transparent="slider",
        )

        mesh.plot_3d_slicer(
            actvMap * sumMap * mrecSum,
            aspect="equal",
            zslice=30,
            pcolor_opts={"cmap": "inferno_r"},
            transparent="slider",
        )


if __name__ == "__main__":
    run()
    plt.show()

Total running time of the script: (0 minutes 31.721 seconds)

Estimated memory usage: 328 MB

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