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.

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Running inversion with SimPEG v0.23.1.dev1+g60e0c3a5d

                    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  5.80e+05  9.11e+06  4.27e-04  9.11e+06    6.29e+01      0
   1  2.90e+05  1.31e+05  7.53e-02  1.53e+05    3.48e+01      0
   2  1.45e+05  4.69e+04  2.30e-01  8.02e+04    4.83e+01      0
   3  7.25e+04  2.00e+04  3.59e-01  4.60e+04    5.49e+01      0   Skip BFGS
   4  3.63e+04  6.49e+03  4.73e-01  2.36e+04    4.88e+01      0   Skip BFGS
   5  1.81e+04  4.62e+03  5.08e-01  1.38e+04    3.55e+01      1   Skip BFGS
   6  9.07e+03  9.06e+02  5.99e-01  6.34e+03    4.80e+01      0
   7  4.53e+03  8.93e+02  6.00e-01  3.61e+03    5.24e+01      4   Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 0.010243003398850706
irls_threshold 0.01204888380086274
   8  4.53e+03  3.96e+02  8.67e-01  4.33e+03    1.07e+01      0
   9  4.53e+03  4.14e+02  9.22e-01  4.59e+03    3.51e+01      3
  10  2.79e+03  5.01e+02  9.82e-01  3.24e+03    5.77e+01      0
  11  1.71e+03  5.01e+02  1.00e+00  2.22e+03    5.76e+01     13   Skip BFGS
  12  1.07e+03  4.86e+02  1.01e+00  1.57e+03    3.41e+01      2
  13  1.30e+03  3.58e+02  1.07e+00  1.74e+03    3.26e+01      0
Minimum decrease in regularization. End of IRLS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 9.1133e+05
1 : |xc-x_last| = 8.0546e-04 <= tolX*(1+|x0|) = 1.0075e-01
0 : |proj(x-g)-x|    = 3.2598e+01 <= tolG          = 1.0000e-03
0 : |proj(x-g)-x|    = 3.2598e+01 <= 1e3*eps       = 1.0000e-03
0 : maxIter   =     100    <= iter          =     14
------------------------- 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()

Estimated memory usage: 327 MB