DCIP with PyGIMLi (Synthetic example)#

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Using the DCIP (Direct Current, Induced Polarization) solver provided by PyGIMLi, we use different cofi solvers to solve the corresponding inverse problem.

Note: This notebook is adapted from a PyGIMLi example: Naive complex-valued electrical inversion

The key difference between ERT and DCIP as implemented in PyGIMLi is that for DCIP resistivties are expressed as complex numbers with the real part representing the resistivity and the phase angle presenting the chargeability. This means that entries into the model vector and the data vector are complex numbers and that DCIP inversions using PyGIMLI rely on the induced polarization field measurements being expressed in the frequency domain.

While numpy.linalg.solve is able to call the appropriate Lapack subroutine for a complex linear system cgesv or zcgesv, other solvers typically expect the model vector and data vector to be real. This means that the complex system of equation needs to be transformed into a real system of equations. Such a transformation needs to be accounted for in the user provided functions for the objective function, Hessian and gradient; care must also be taken when transforming the data covariance matrix.

The linear equation $ A x =b $ with the elements of \(A\), \(b\) and \(x\) being complex numbers can be rewritten using real numbers as follows

\[\begin{split}\begin{pmatrix}A^r & -A^c \\A^c & A^r \end{pmatrix} \begin{pmatrix} x^r \\ x^c \end{pmatrix} = \begin{pmatrix} b^r \\ b^c \end{pmatrix},\end{split}\]

with \(b=( b_1^r+b_1^c i, b_2^r+b_2^c i,...,b_n^r+b_n^c i)\) being rewritten as \((b^r,b^c)\) with \(b^r=(b_1^r,b_2^r,...,b_n^r)\) and \(b^c=(b_1^c,b_2^c,...,b_n^c)\) and analogus reordering for \(A\) and \(x\).

See https://ijpam.eu/contents/2012-76-1/11/11.pdf for more details.

1. Import modules#

# -------------------------------------------------------- #
#                                                          #
#     Uncomment below to set up environment on "colab"     #
#                                                          #
# -------------------------------------------------------- #

# !pip install -U cofi

# !pip install -q condacolab
# import condacolab
# condacolab.install()
# !mamba install -c gimli pygimli=1.3

We will need the following packages:

  • numpy for matrices and matrix-related functions

  • matplotlib for plotting

  • pygimli for forward modelling of the problem

  • cofi for accessing different inference solvers

import numpy as np
import matplotlib.pyplot as plt
import pygimli
import cofi

np.random.seed(42)

Below we define a set of utility functions that help define the problem, generating data and making plots. Feel free to skip reading the details of these utility functions and come back later if you want.

1.1. Helper functions for complex numbers#

def rho_phi_to_complex(rho, phi):      # rho * e^(phi * i)
    return pygimli.utils.toComplex(rho, phi)

def rho_phi_from_complex(complx):      # |complx|, arctan(complx.imag, complx.real)
    return np.abs(complx), np.arctan2(complx.imag, complx.real)

def complex_to_real(complx):           # complx vector of size n -> size 2n
    return pygimli.utils.squeezeComplex(complx)

def complex_from_real(real):           # real vector of size n -> size n/2
    return pygimli.utils.toComplex(real)

1.2. Helper functions for PyGIMLi modelling#

# Utility Functions
x_inv_start = -2
x_inv_stop = 52
y_inv_start = -20
y_inv_stop = 0

def survey_scheme(start=0, stop=50, num=51, schemeName="dd"):
    scheme = pygimli.physics.ert.createData(elecs=np.linspace(start=start, stop=stop, num=num),schemeName=schemeName)
    return scheme

def model_true(
    scheme,
    start=[-55, 0],
    end=[105, -80],
    anomalies_pos=[[10,-7],[40,-7]],
    anomalies_rad=[5,5],
    rhomap=[[1, rho_phi_to_complex(100, 0 / 1000)],
          # Magnitude: 50 ohm m, Phase: -50 mrad
          [2, rho_phi_to_complex(50, 0 / 1000)],
          [3, rho_phi_to_complex(100, -50 / 1000)],]
    ):
    world = pygimli.meshtools.createWorld(start=start, end=end, worldMarker=True)
    for s in scheme.sensors():          # local refinement
        world.createNode(s + [0.0, -0.1])
    geom = world
    for i, (pos, rad) in enumerate(zip(anomalies_pos, anomalies_rad)):
        anomaly = pygimli.meshtools.createCircle(pos=pos, radius=rad, marker=i+2)
        geom += anomaly
    mesh = pygimli.meshtools.createMesh(geom, quality=33)
    return mesh, rhomap

def ert_simulate(mesh, scheme, rhomap, noise_level=1, noise_abs=1e-6):
    pg_data = pygimli.physics.ert.simulate(mesh, scheme=scheme, res=rhomap, noiseLevel=noise_level,
                        noise_abs=noise_abs, seed=42)
    # data.remove(data["rhoa"] < 0)
    data_complex = rho_phi_to_complex(pg_data["rhoa"].array(), pg_data["phia"].array())
    data_log_complex = np.log(data_complex)
    return pg_data, data_complex, data_log_complex

def ert_manager(pg_data, verbose=False):
    return pygimli.physics.ert.ERTManager(pg_data, verbose=verbose, useBert=True)

def inversion_mesh(ert_mgr):
    inv_mesh = ert_mgr.createMesh(ert_mgr.data)
    # print("model size", inv_mesh.cellCount())   # 1031
    ert_mgr.setMesh(inv_mesh)
    return inv_mesh

def ert_forward_operator(ert_mgr, pg_data, inv_mesh):
    forward_oprt = ert_mgr.fop
    forward_oprt.setComplex(True)
    forward_oprt.setData(pg_data)
    forward_oprt.setMesh(inv_mesh, ignoreRegionManager=True)
    return forward_oprt

def reg_matrix(forward_oprt):
    region_manager = forward_oprt.regionManager()
    region_manager.setConstraintType(2)
    Wm = pygimli.matrix.SparseMapMatrix()
    region_manager.fillConstraints(Wm)
    Wm = pygimli.utils.sparseMatrix2coo(Wm)
    return Wm

def starting_model(data, inv_mesh, rho_val=None, phi_val=None):
    rho_start = np.median(data["rhoa"]) if rho_val is None else rho_val
    phi_start = np.median(data["phia"]) if phi_val is None else phi_val
    start_model_val = rho_phi_to_complex(rho_start, phi_start)
    start_model_complex = np.ones(inv_mesh.cellCount()) * start_model_val
    start_model_log_complex = np.log(start_model_complex)
    start_model_log_real = complex_to_real(start_model_log_complex)
    return start_model_complex, start_model_log_complex, start_model_log_real

def model_vector(rhomap, mesh):
    return pygimli.solver.parseArgToArray(rhomap, mesh.cellCount(), mesh).array()

1.3. Helper functions for plotting#

def plot_model(mesh, model_complex, title):
    rho, phi = rho_phi_from_complex(model_complex)
    fig, axes = plt.subplots(1,2,figsize=(10,3))
    pygimli.show(mesh, data=rho, label=r"$\Omega m$", ax=axes[0])
    axes[0].set_xlim(x_inv_start, x_inv_stop)
    axes[0].set_ylim(y_inv_start, y_inv_stop)
    axes[0].set_title("Resistivity")
    pygimli.show(mesh, data=phi * 1000, label=r"mrad", ax=axes[1])
    axes[1].set_xlim(x_inv_start, x_inv_stop)
    axes[1].set_ylim(y_inv_start, y_inv_stop)
    axes[1].set_title("Chargeability")
    fig.suptitle(title)

def plot_data(pg_data, data_complex, title):
    rho, phi = rho_phi_from_complex(data_complex)
    fig, axes = plt.subplots(1,2,figsize=(10,4))
    pygimli.physics.ert.showERTData(pg_data, vals=rho, label=r"$\Omega$m", ax=axes[0])
    axes[0].set_title("Apparent Resistivity")
    pygimli.physics.ert.showERTData(pg_data, vals=phi*1000, label=r"mrad", ax=axes[1])
    axes[1].set_title("Apparent Chargeability")
    fig.suptitle(title)

2. Define the problem#

We first define the true model, the survey and map it on a computational mesh designed for the survey and true anomaly.

2.1. True model#

# PyGIMLi - define measuring scheme, geometry, forward mesh and true model
scheme = survey_scheme()
mesh, rhomap = model_true(scheme)

# plot the true model
plot_model(mesh, model_vector(rhomap, mesh), "True model")
True model, Resistivity, Chargeability

2.2. Generate synthetic data#

Generate the synthetic data as a container with all the necessary information for plotting:

pg_data, data_complex, data_log_complex = ert_simulate(mesh, scheme, rhomap)

plot_data(pg_data, data_complex, "(Synthetic) Data Observatons")
(Synthetic) Data Observatons, Apparent Resistivity, Apparent Chargeability
relativeError set to a value > 0.5 .. assuming this is a percentage Error level dividing them by 100
Data error estimate (min:max)  0.010018232527676921 : 0.39689542284224183
Data IP abs error estimate (min:max)  7.507792421169821e-10 : 0.00022318579899318784

2.3. ERTManager#

Further, we create a pygimli.ert.ERTManager instance to keep record of problem-specific information like the inversion mesh, and to perform forward operation for the inversion solvers.

# create PyGIMLi's ERT manager
ert_mgr = ert_manager(pg_data)

2.4. Inversion mesh#

The inversion can use a different mesh and the mesh to be used should know nothing about the mesh that was designed based on the true model. Here we first use a triangular mesh for the inversion, which makes the problem underdetermined.

inv_mesh = inversion_mesh(ert_mgr)

ax = pygimli.show(inv_mesh, showMesh=True, markers=True, colorBar=False)
ax[0].set_title("Mesh used for inversion")
Mesh used for inversion
Text(0.5, 1.0, 'Mesh used for inversion')

2.5. Forward operator, regularization matrix#

With the inversion mesh created, we now define a starting model, forward operator and weighting matrix for regularization using PyGIMLi.

Our model will be in log space when we perform inversion (for numerical stability purposes).

# PyGIMLi's forward operator (ERTModelling)
forward_oprt = ert_forward_operator(ert_mgr, scheme, inv_mesh)

# extract regularization matrix
Wm = reg_matrix(forward_oprt)

# initialise a starting model for inversion
start_model, start_model_log, start_model_log_real = starting_model(pg_data, ert_mgr.paraDomain)
plot_model(ert_mgr.paraDomain, start_model, "Starting model")
Starting model, Resistivity, Chargeability

2.6. Utility functions to pass to CoFI#

CoFI and other inference packages require a set of functions that provide the misfit, the jacobian the residual within the case of scipy standardised interfaces. All these functions are defined below as additional utility functions, so feel free to read them into details if you want to understand more. These functions are:

  • get_response

  • get_jacobian

  • get_residuals

  • get_data_misfit

  • get_regularization

  • get_gradient

  • get_hessian

# Utility Functions (additional)

def _ensure_numpy(model):
    if "torch.Tensor" in str(type(model)):
        model = model.cpu().detach().numpy()
    return model

# model_log_complex -> data_log_complex
def get_response(model_log_complex, fop):
    model_complex = np.exp(model_log_complex)
    model_real = complex_to_real(model_complex)
    model_real = _ensure_numpy(model_real)
    data_real = np.array(fop.response(model_real))
    data_complex = complex_from_real(data_real)
    data_log_complex = np.log(data_complex)
    return data_log_complex

# model_log_complex -> J_log_log_complex
def get_jacobian(model_log_complex, fop):
    model_complex = np.exp(model_log_complex)
    model_real = complex_to_real(model_complex)
    model_real = _ensure_numpy(model_real)
    J_block = fop.createJacobian(model_real)
    J_real = np.array(J_block.mat(0))
    J_imag = np.array(J_block.mat(1))
    J_complex = J_real + 1j * J_imag
    data_log_complex = get_response(model_log_complex, fop)
    data_complex = np.exp(data_log_complex)
    J_log_log_complex = J_complex / data_complex[:,np.newaxis] * model_complex[np.newaxis,:]
    return J_log_log_complex

# model_log_complex -> res_data_log_complex
def get_residuals(model_log_complex, data_log_complex, fop):
    synth_data_log_complex = get_response(model_log_complex, fop)
    return data_log_complex - synth_data_log_complex

# model_log_real -> obj_log_real
def get_objective(model_log_real, data_log_complex, fop, lamda, Wm):
    # convert model_log_real into complex numbers
    model_log_complex = complex_from_real(model_log_real)
    # calculate data misfit
    res_log_complex = get_residuals(model_log_complex, data_log_complex, fop)
    data_misfit = res_log_complex.conj().dot(res_log_complex)
    # calculate regularization term
    weighted_model_log_real = Wm.dot(model_log_complex)
    reg = lamda * weighted_model_log_real.conj().dot(weighted_model_log_real)
    # sum up
    result = np.abs(data_misfit + reg)
    return result

# model_log_real -> grad_log_real
def get_gradient(model_log_real, data_log_complex, fop, lamda, Wm):
    # convert model_log_real into complex numbers
    model_log_complex = complex_from_real(model_log_real)
    # calculate gradient for data misfit
    res = get_residuals(model_log_complex, data_log_complex, fop)
    jac = get_jacobian(model_log_complex, fop)
    data_misfit_grad = - jac.conj().T.dot(res)
    # calculate gradient for regularization term
    reg_grad = lamda * Wm.T.dot(Wm).dot(model_log_complex)
    # sum up
    grad_complex = data_misfit_grad + reg_grad
    grad_real = complex_to_real(grad_complex)
    return grad_real

# model_log_real -> hess_log_real
def get_hessian(model_log_real, data_log_complex, fop, lamda, Wm):
    # convert model_log_real into complex numbers
    model_log_complex = complex_from_real(model_log_real)
    # calculate hessian for data misfit
    res = get_residuals(model_log_complex, data_log_complex, fop)
    jac = get_jacobian(model_log_complex, fop)
    data_misfit_hessian = jac.conj().T.dot(jac)
    # calculate hessian for regularization term
    reg_hessian = lamda * Wm.T.dot(Wm)
    # sum up
    hessian_complex = data_misfit_hessian + reg_hessian
    nparams = len(model_log_complex)
    hessian_real = np.zeros((2*nparams, 2*nparams))
    hessian_real[:nparams,:nparams] = np.real(hessian_complex)
    hessian_real[:nparams,nparams:] = -np.imag(hessian_complex)
    hessian_real[nparams:,:nparams] = np.imag(hessian_complex)
    hessian_real[nparams:,nparams:] = np.real(hessian_complex)
    return hessian_real

With all the above forward operations set up with PyGIMLi, we now define the problem in cofi by setting the problem information for a BaseProblem object.

# hyperparameters
lamda=0.001

# CoFI - define BaseProblem
dcip_problem = cofi.BaseProblem()
dcip_problem.name = "DC-IP defined through PyGIMLi"
dcip_problem.set_objective(get_objective, args=[data_log_complex, forward_oprt, lamda, Wm])
dcip_problem.set_gradient(get_gradient, args=[data_log_complex, forward_oprt, lamda, Wm])
dcip_problem.set_hessian(get_hessian, args=[data_log_complex, forward_oprt, lamda, Wm])
dcip_problem.set_initial_model(start_model_log_real)

3. Define the inversion options and run#

3.1. SciPy’s optimizer (trust-ncg)#

dcip_problem.suggest_tools();
Based on what you've provided so far, here are possible tools:
{
    "optimization": [
        "scipy.optimize.minimize",
        "torch.optim"
    ],
    "matrix solvers": [
        "cofi.simple_newton"
    ],
    "sampling": []
}

{'optimization': ['scipy.optimize.minimize', 'torch.optim'], 'matrix solvers': ['cofi.simple_newton'], 'sampling': []}
inv_options_scipy = cofi.InversionOptions()
inv_options_scipy.set_tool("scipy.optimize.minimize")
inv_options_scipy.set_params(method="trust-ncg", options={"maxiter":5})
inv_scipy = cofi.Inversion(dcip_problem, inv_options_scipy)
inv_result_scipy = inv_scipy.run()
print(f"\nSolver message: {inv_result_scipy.message}")
Solver message: Maximum number of iterations has been exceeded.
model_scipy = np.exp(complex_from_real(inv_result_scipy.model))
plot_model(ert_mgr.paraDomain, model_scipy, "Inferred model (scipy's trust-ncg)")

synth_data_scipy = np.exp(get_response(np.log(model_scipy), forward_oprt))
plot_data(pg_data, synth_data_scipy, "Inferred model produced data")
  • Inferred model (scipy's trust-ncg), Resistivity, Chargeability
  • Inferred model produced data, Apparent Resistivity, Apparent Chargeability

3.2. PyTorch’s optimizer (RAdam)#

inv_options_torch = cofi.InversionOptions()
inv_options_torch.set_tool("torch.optim")
inv_options_torch.set_params(algorithm="RAdam", lr=0.05, num_iterations=20)
inv_torch = cofi.Inversion(dcip_problem, inv_options_torch)
inv_result_torch = inv_torch.run()
/home/jiawen/cofi/src/cofi/solvers/base_solver.py:267: UserWarning: the following options are defined but not in parameter list for the chosen tool: ['lr']
  warnings.warn(
Iteration #0, objective value: 40.95771199605861
Iteration #1, objective value: 33.006761054365505
Iteration #2, objective value: 27.517003420367388
Iteration #3, objective value: 23.856374111760857
Iteration #4, objective value: 21.50039198382136
Iteration #5, objective value: 20.03191354052629
Iteration #6, objective value: 19.94374499737532
Iteration #7, objective value: 19.827200872779063
Iteration #8, objective value: 19.687514187805156
Iteration #9, objective value: 19.529728014807965
Iteration #10, objective value: 19.35862270111142
Iteration #11, objective value: 19.178021476356154
Iteration #12, objective value: 18.990944837953872
Iteration #13, objective value: 18.800752066476278
Iteration #14, objective value: 18.61164377300838
Iteration #15, objective value: 18.428027398836804
Iteration #16, objective value: 18.253580046609493
Iteration #17, objective value: 18.090719054270597
Iteration #18, objective value: 17.94081235117483
Iteration #19, objective value: 17.804786009588636
model_torch = np.exp(complex_from_real(inv_result_torch.model))
plot_model(ert_mgr.paraDomain, model_torch, "Inferred model (torch.optim.RAdam)")

synth_data_torch = np.exp(get_response(np.log(model_torch), forward_oprt))
plot_data(pg_data, synth_data_torch, "Inferred model produced data")
  • Inferred model (torch.optim.RAdam), Resistivity, Chargeability
  • Inferred model produced data, Apparent Resistivity, Apparent Chargeability

Watermark#

watermark_list = ["cofi", "numpy", "scipy", "pygimli", "torch", "matplotlib"]
for pkg in watermark_list:
    pkg_var = __import__(pkg)
    print(pkg, getattr(pkg_var, "__version__"))
cofi 0.1.2.dev22
numpy 1.21.6
scipy 1.9.1
pygimli 1.2.6
torch 1.12.1+cu102
matplotlib 3.5.3

sphinx_gallery_thumbnail_number = -1

Total running time of the script: ( 3 minutes 4.958 seconds)

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