Straight Ray with Volume Data Misfit Term

Based on the SEG abstract Heagy, Cockett and Oldenburg, 2014.

Heagy, L. J., Cockett, A. R., & Oldenburg, D. W. (2014, August 5). Parametrized Inversion Framework for Proppant Volume in a Hydraulically Fractured Reservoir. SEG Technical Program Expanded Abstracts 2014. Society of Exploration Geophysicists. doi:10.1190/segam2014-1639.1

This example is a simple joint inversion that consists of a

• data misfit for the tomography problem

• data misfit for the volume of the inclusions (uses the effective medium theory mapping)

• model regularization

• 
• 
• 
• 

True Volume: 0.6240000000000001
/home/vsts/work/1/s/examples/20-published/plot_tomo_joint_with_volume.py:157: FutureWarning: The defaults for ProjectedGNCG will change in SimPEG 0.26.0. If you want to maintain the previous behavior, explicitly set 'cg_atol=1E-3' and 'cg_rtol=0.0'.
  opt = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)

Running inversion with SimPEG v0.25.2
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  2.50e-01  8.50e+04  0.00e+00  8.50e+04                         0           inf          inf
   1  2.50e-01  3.95e+03  9.97e-01  3.95e+03    2.48e-01      0      5        4.21e-03     8.96e+00
/home/vsts/work/1/s/simpeg/maps/_property_maps.py:1505: UserWarning: Maximum number of iterations reached
  return self._sc2phaseEMTSpheroidstransform(m)
   2  2.50e-01  6.07e+02  8.41e-01  6.07e+02    3.28e+01      0      5        4.26e-03     1.74e-01
   3  2.50e-01  1.23e+02  7.27e-01  1.23e+02    2.93e+01      1      5        7.98e-01     1.59e+02
   4  2.50e-01  4.24e+01  7.03e-01  4.25e+01    2.17e+01      1      5        4.43e-01     5.09e+01
   5  2.50e-01  2.57e+01  6.98e-01  2.59e+01    1.24e+01      0      5        5.01e-01     3.68e+01
   6  2.50e-01  1.97e+01  7.00e-01  1.99e+01    1.19e+01      0      5        7.09e-01     3.17e+01
   7  2.50e-01  1.63e+01  6.99e-01  1.65e+01    1.04e+01      0      5        1.32e+00     3.12e+01
   8  2.50e-01  1.24e+01  6.99e-01  1.26e+01    1.01e+01      0      5        4.12e-01     1.83e+01
   9  2.50e-01  9.70e+00  7.01e-01  9.88e+00    8.83e+00      0      5        1.11e+00     2.52e+01
  10  2.50e-01  8.45e+00  7.01e-01  8.63e+00    7.73e+00      0      5        8.12e-01     1.95e+01
  11  2.50e-01  6.93e+00  7.02e-01  7.11e+00    7.68e+00      0      5        4.60e-01     1.46e+01
  12  2.50e-01  5.99e+00  7.03e-01  6.16e+00    6.75e+00      0      5        9.52e-01     1.81e+01
  13  2.50e-01  4.65e+00  7.05e-01  4.82e+00    6.46e+00      0      5        6.97e-01     1.66e+01
  14  2.50e-01  3.89e+00  7.04e-01  4.07e+00    5.64e+00      0      5        8.62e-01     1.60e+01   Skip BFGS
  15  2.50e-01  3.60e+00  7.04e-01  3.78e+00    5.32e+00      0      5        9.07e-01     1.39e+01
------------------------- STOP! -------------------------
1 : |fc-fOld| = 2.8982e-01 <= tolF*(1+|f0|) = 8.4963e+03
1 : |xc-x_last| = 5.7209e-02 <= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 4.6891e+00 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 4.6891e+00 <= 1e3*eps       = 1.0000e-02
1 : maxIter   =      15    <= iter          =     15
------------------------- DONE! -------------------------

Total recovered volume (no vol misfit term in inversion): 0.024598336889261475
/home/vsts/work/1/s/examples/20-published/plot_tomo_joint_with_volume.py:172: FutureWarning: The defaults for ProjectedGNCG will change in SimPEG 0.26.0. If you want to maintain the previous behavior, explicitly set 'cg_atol=1E-3' and 'cg_rtol=0.0'.
  opt2 = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)

Running inversion with SimPEG v0.25.2
================================================= Projected GNCG =================================================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS   iter_CG   CG |Ax-b|/|b|  CG |Ax-b|   Comment
-----------------------------------------------------------------------------------------------------------------
   0  2.50e-01  1.20e+05  0.00e+00  1.20e+05                         0           inf          inf
   1  2.50e-01  7.77e+03  8.53e-01  7.77e+03    2.22e+01      0      5        5.16e-02     7.58e+01
   2  2.50e-01  7.74e+03  8.39e-01  7.74e+03    2.69e+01      5      5        3.12e-01     1.62e+02
   3  2.50e-01  7.67e+03  8.25e-01  7.67e+03    2.64e+01      4      5        5.11e-01     2.13e+02   Skip BFGS
------------------------------------------------------------------
0 :    ft     = 7.6713e+03 <= alp*descent     = 7.6693e+03
1 : maxIterLS =      10    <= iterLS          =     10
------------------------- End Linesearch -------------------------
The linesearch got broken. Boo.

Total volume (vol misfit term in inversion): 0.026410253704576843

import numpy as np
import scipy.sparse as sp
import matplotlib.pyplot as plt

from simpeg.seismic import straight_ray_tomography as tomo
import discretize
from simpeg import (
    maps,
    utils,
    regularization,
    optimization,
    inverse_problem,
    inversion,
    data_misfit,
    objective_function,
)


class Volume(objective_function.BaseObjectiveFunction):
    r"""
    A regularization on the volume integral of the model

    .. math::

        \phi_v = || \int_V m dV - \text{knownVolume} ||^2
    """

    def __init__(self, mesh, knownVolume=0.0, **kwargs):
        self.mesh = mesh
        self.knownVolume = knownVolume
        super().__init__(**kwargs)

    @property
    def knownVolume(self):
        """known volume"""
        return self._knownVolume

    @knownVolume.setter
    def knownVolume(self, value):
        self._knownVolume = utils.validate_float("knownVolume", value, min_val=0.0)

    def __call__(self, m):
        return (self.estVol(m) - self.knownVolume) ** 2

    def estVol(self, m):
        return np.inner(self.mesh.cell_volumes, m)

    def deriv(self, m):
        # return (self.mesh.cell_volumes * np.inner(self.mesh.cell_volumes, m))
        return (
            2
            * self.mesh.cell_volumes
            * (self.knownVolume - np.inner(self.mesh.cell_volumes, m))
        )  # factor of 2 from deriv of ||estVol - knownVol||^2

    def deriv2(self, m, v=None):
        if v is not None:
            return 2 * utils.mkvc(
                self.mesh.cell_volumes * np.inner(self.mesh.cell_volumes, v)
            )
        else:
            # TODO: this is inefficent. It is a fully dense matrix
            return 2 * sp.csc_matrix(
                np.outer(self.mesh.cell_volumes, self.mesh.cell_volumes)
            )


def run(plotIt=True):
    nC = 40
    de = 1.0
    h = np.ones(nC) * de / nC
    M = discretize.TensorMesh([h, h])

    y = np.linspace(M.cell_centers_y[0], M.cell_centers_x[-1], int(np.floor(nC / 4)))
    rlocs = np.c_[0 * y + M.cell_centers_x[-1], y]
    rx = tomo.Rx(rlocs)

    source_list = [
        tomo.Src(location=np.r_[M.cell_centers_x[0], yi], receiver_list=[rx])
        for yi in y
    ]

    # phi model
    phi0 = 0
    phi1 = 0.65
    phitrue = utils.model_builder.create_block_in_wholespace(
        M.gridCC, [0.4, 0.6], [0.6, 0.4], [phi1, phi0]
    )

    knownVolume = np.sum(phitrue * M.cell_volumes)
    print("True Volume: {}".format(knownVolume))

    # Set up true conductivity model and plot the model transform
    sigma0 = np.exp(1)
    sigma1 = 1e4

    if plotIt:
        fig, ax = plt.subplots(1, 1)
        sigmaMapTest = maps.SelfConsistentEffectiveMedium(
            nP=1000, sigma0=sigma0, sigma1=sigma1, rel_tol=1e-1, maxIter=150
        )
        testphis = np.linspace(0.0, 1.0, 1000)

        sigetest = sigmaMapTest * testphis
        ax.semilogy(testphis, sigetest)
        ax.set_title("Model Transform")
        ax.set_xlabel(r"$\varphi$")
        ax.set_ylabel(r"$\sigma$")

    sigmaMap = maps.SelfConsistentEffectiveMedium(M, sigma0=sigma0, sigma1=sigma1)

    # scale the slowness so it is on a ~linear scale
    slownessMap = maps.LogMap(M) * sigmaMap

    # set up the problem and survey
    survey = tomo.Survey(source_list)
    problem = tomo.Simulation(M, survey=survey, slownessMap=slownessMap)

    if plotIt:
        _, ax = plt.subplots(1, 1)
        cb = plt.colorbar(M.plot_image(phitrue, ax=ax)[0], ax=ax)
        survey.plot(ax=ax)
        cb.set_label(r"$\varphi$")

    # get observed data
    data = problem.make_synthetic_data(phitrue, relative_error=0.03, add_noise=True)
    dpred = problem.dpred(np.zeros(M.nC))

    # objective function pieces
    reg = regularization.WeightedLeastSquares(M)
    dmis = data_misfit.L2DataMisfit(simulation=problem, data=data)
    dmisVol = Volume(mesh=M, knownVolume=knownVolume)
    beta = 0.25
    maxIter = 15

    # without the volume regularization
    opt = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
    opt.remember("xc")
    invProb = inverse_problem.BaseInvProblem(dmis, reg, opt, beta=beta)
    inv = inversion.BaseInversion(invProb)

    mopt1 = inv.run(np.zeros(M.nC) + 1e-16)
    print(
        "\nTotal recovered volume (no vol misfit term in inversion): "
        "{}".format(dmisVol(mopt1))
    )

    # with the volume regularization
    vol_multiplier = 9e4
    reg2 = reg
    dmis2 = dmis + vol_multiplier * dmisVol
    opt2 = optimization.ProjectedGNCG(maxIter=maxIter, lower=0.0, upper=1.0)
    opt2.remember("xc")
    invProb2 = inverse_problem.BaseInvProblem(dmis2, reg2, opt2, beta=beta)
    inv2 = inversion.BaseInversion(invProb2)

    mopt2 = inv2.run(np.zeros(M.nC) + 1e-16)
    print("\nTotal volume (vol misfit term in inversion): {}".format(dmisVol(mopt2)))

    # plot results

    if plotIt:
        fig, ax = plt.subplots(1, 1)
        ax.plot(data.dobs)
        ax.plot(dpred)
        ax.plot(problem.dpred(mopt1), "o")
        ax.plot(problem.dpred(mopt2), "s")
        ax.legend(["dobs", "dpred0", "dpred w/o Vol", "dpred with Vol"])

        fig, ax = plt.subplots(1, 3, figsize=(16, 4))
        im0 = M.plot_image(phitrue, ax=ax[0])[0]
        im1 = M.plot_image(mopt1, ax=ax[1])[0]
        im2 = M.plot_image(mopt2, ax=ax[2])[0]

        for im in [im0, im1, im2]:
            im.set_clim([0.0, phi1])

        plt.colorbar(im0, ax=ax[0])
        plt.colorbar(im1, ax=ax[1])
        plt.colorbar(im2, ax=ax[2])

        ax[0].set_title("true, vol: {:1.3e}".format(knownVolume))
        ax[1].set_title(
            "recovered(no Volume term), vol: {:1.3e} ".format(dmisVol(mopt1))
        )
        ax[2].set_title(
            "recovered(with Volume term), vol: {:1.3e} ".format(dmisVol(mopt2))
        )

        plt.tight_layout()


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

Estimated memory usage: 325 MB