# 3D Least-Squares Inversion of DC and IP Data#

Here we invert 5 lines of DC and IP data to recover both an electrical conductivity and a chargeability model. We formulate the corresponding inverse problems as least-squares optimization problems. For this tutorial, we focus on the following:

• Generating a mesh based on survey geometry

• Including surface topography

• Defining the inverse problem (data misfit, regularization, directives)

• Applying sensitivity weighting

• Plotting the recovered model and data misfit

The DC data are measured voltages normalized by the source current in V/A and the IP data are defined as apparent chargeabilities and V/V.

## Import Modules#

import os
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import tarfile

from discretize import TreeMesh
from discretize.utils import refine_tree_xyz, active_from_xyz

from SimPEG.utils import model_builder
from SimPEG import (
maps,
data_misfit,
regularization,
optimization,
inverse_problem,
inversion,
directives,
utils,
)
from SimPEG.electromagnetics.static import resistivity as dc
from SimPEG.electromagnetics.static import induced_polarization as ip
from SimPEG.electromagnetics.static.utils.static_utils import (
apparent_resistivity_from_voltage,
)

# To plot DC/IP data in 3D, the user must have the plotly package
try:
import plotly
from SimPEG.electromagnetics.static.utils.static_utils import plot_3d_pseudosection

has_plotly = True
except ImportError:
has_plotly = False
pass

try:
from pymatsolver import Pardiso as Solver
except ImportError:
from SimPEG import SolverLU as Solver

mpl.rcParams.update({"font.size": 16})

# sphinx_gallery_thumbnail_number = 7


## Define File Names#

Here we provide the file paths to assets we need to run the inversion. The path to the true model conductivity and chargeability models are also provided for comparison with the inversion results. These files are stored as a tar-file on our google cloud bucket: “https://storage.googleapis.com/simpeg/doc-assets/dcip3d.tar.gz

# storage bucket where we have the data

# unzip the tarfile
tar.extractall()
tar.close()

# path to the directory containing our data

# files to work with
topo_filename = dir_path + "topo_xyz.txt"
dc_data_filename = dir_path + "dc_data.xyz"
ip_data_filename = dir_path + "ip_data.xyz"

Downloading https://storage.googleapis.com/simpeg/doc-assets/dcip3d.tar.gz
saved to: /home/vsts/work/1/s/tutorials/06-ip/dcip3d.tar.gz


Here we load the observed data and topography.

topo_xyz = np.loadtxt(str(topo_filename))

dc_data_filename,
"volt",
is_surface_data=False,
)

ip_data_filename,
"apparent_chargeability",
is_surface_data=False,
)


## Plot Observed DC Data in Pseudosection#

Here we plot the observed DC data in 3D pseudosection. To use this utility, you must have Python’s plotly package. Here, we represent the DC data as apparent conductivities.

# Convert predicted data to apparent conductivities
apparent_conductivity = 1 / apparent_resistivity_from_voltage(
dc_data.survey,
dc_data.dobs,
)

if has_plotly:
# Plot DC Data
fig = plot_3d_pseudosection(
dc_data.survey, apparent_conductivity, scale="log", units="S/m"
)

fig.update_layout(
title_text="Apparent Conductivity",
title_x=0.5,
title_font_size=24,
width=650,
height=500,
scene_camera=dict(
center=dict(x=0, y=0, z=-0.4), eye=dict(x=1.5, y=-1.5, z=1.8)
),
)

plotly.io.show(fig)

else:
print("INSTALL 'PLOTLY' TO VISUALIZE 3D PSEUDOSECTIONS")


## Plot Observed IP Data in Pseudosection#

Here we plot the observed IP data in 3D pseudosection. To use this utility, you must have Python’s plotly package. Here, we represent the IP data as apparent chargeabilities.

if has_plotly:
# Plot IP Data
fig = plot_3d_pseudosection(
ip_data.survey,
ip_data.dobs,
scale="linear",
units="V/V",
vlim=[0, np.max(ip_data.dobs)],
marker_opts={"colorscale": "plasma"},
)

fig.update_layout(
title_text="Apparent Chargeability",
title_x=0.5,
title_font_size=24,
width=650,
height=500,
scene_camera=dict(
center=dict(x=0, y=0, z=-0.4), eye=dict(x=1.5, y=-1.5, z=1.8)
),
)

plotly.io.show(fig)

else:
print("INSTALL 'PLOTLY' TO VISUALIZE 3D PSEUDOSECTIONS")


## Assign Uncertainties#

Inversion with SimPEG requires that we define the uncertainties on our data. This represents our estimate of the standard deviation of the noise in our data. For DC data, the uncertainties are 10% of the absolute value. For IP data, the uncertainties are 5e-3 V/V.

dc_data.standard_deviation = 0.1 * np.abs(dc_data.dobs)
ip_data.standard_deviation = 5e-3 * np.ones_like(ip_data.dobs)


## Create Tree Mesh#

Here, we create the Tree mesh that will be used to invert both DC resistivity and IP data.

dh = 25.0  # base cell width
dom_width_x = 6000.0  # domain width x
dom_width_y = 6000.0  # domain width y
dom_width_z = 4000.0  # domain width z
nbcx = 2 ** int(np.round(np.log(dom_width_x / dh) / np.log(2.0)))  # num. base cells x
nbcy = 2 ** int(np.round(np.log(dom_width_y / dh) / np.log(2.0)))  # num. base cells y
nbcz = 2 ** int(np.round(np.log(dom_width_z / dh) / np.log(2.0)))  # num. base cells z

# Define the base mesh
hx = [(dh, nbcx)]
hy = [(dh, nbcy)]
hz = [(dh, nbcz)]
mesh = TreeMesh([hx, hy, hz], x0="CCN")

# Mesh refinement based on topography
k = np.sqrt(np.sum(topo_xyz[:, 0:2] ** 2, axis=1)) < 1200
mesh = refine_tree_xyz(
mesh, topo_xyz[k, :], octree_levels=[0, 6, 8], method="surface", finalize=False
)

# Mesh refinement near sources and receivers.
electrode_locations = np.r_[
dc_data.survey.locations_a,
dc_data.survey.locations_b,
dc_data.survey.locations_m,
dc_data.survey.locations_n,
]
unique_locations = np.unique(electrode_locations, axis=0)
mesh = refine_tree_xyz(
mesh, unique_locations, octree_levels=[4, 6, 4], method="radial", finalize=False
)

# Finalize the mesh
mesh.finalize()


## Project Electrodes to Discretized Topography#

It is important that electrodes are not modeled as being in the air. Even if the electrodes are properly located along surface topography, they may lie above the discretized topography. This step is carried out to ensure all electrodes lie on the discretized surface.

# Find cells that lie below surface topography
ind_active = active_from_xyz(mesh, topo_xyz)

# Extract survey from data object
dc_survey = dc_data.survey
ip_survey = ip_data.survey

# Shift electrodes to the surface of discretized topography
dc_survey.drape_electrodes_on_topography(mesh, ind_active, option="top")
ip_survey.drape_electrodes_on_topography(mesh, ind_active, option="top")

# Reset survey in data object
dc_data.survey = dc_survey
ip_data.survey = ip_survey


## Starting/Reference Model and Mapping on OcTree Mesh#

Here, we create starting and/or reference models for the DC inversion as well as the mapping from the model space to the active cells. Starting and reference models can be a constant background value or contain a-priori structures. Here, the starting model is the natural log of 0.01 S/m.

# Define conductivity model in S/m (or resistivity model in Ohm m)
air_conductivity = np.log(1e-8)
background_conductivity = np.log(1e-2)

# Define the mapping from active cells to the entire domain
active_map = maps.InjectActiveCells(mesh, ind_active, np.exp(air_conductivity))
nC = int(ind_active.sum())

# Define the mapping from the model to the conductivity of the entire domain
conductivity_map = active_map * maps.ExpMap()

# Define starting model
starting_conductivity_model = background_conductivity * np.ones(nC)


## Define the Physics of the DC Simulation#

Here, we define the physics of the DC resistivity simulation.

dc_simulation = dc.Simulation3DNodal(
mesh, survey=dc_survey, sigmaMap=conductivity_map, solver=Solver, storeJ=True
)


## Define DC Inverse Problem#

The inverse problem is defined by 3 things:

1. Data Misfit: a measure of how well our recovered model explains the field data

2. Regularization: constraints placed on the recovered model and a priori information

3. Optimization: the numerical approach used to solve the inverse problem

# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# Within the data misfit, the residual between predicted and observed data are
# normalized by the data's standard deviation.
dc_data_misfit = data_misfit.L2DataMisfit(data=dc_data, simulation=dc_simulation)

# Define the regularization (model objective function)
dc_regularization = regularization.WeightedLeastSquares(
mesh,
indActive=ind_active,
reference_model=starting_conductivity_model,
)

dc_regularization.reference_model_in_smooth = (
True  # Include reference model in smoothness
)

# Define how the optimization problem is solved.
dc_optimization = optimization.InexactGaussNewton(maxIter=15, maxIterCG=30, tolCG=1e-2)

# Here we define the inverse problem that is to be solved
dc_inverse_problem = inverse_problem.BaseInvProblem(
dc_data_misfit, dc_regularization, dc_optimization
)


## Define DC Inversion Directives#

Here we define any directives that are carried out during the inversion. This includes the cooling schedule for the trade-off parameter (beta), stopping criteria for the inversion and saving inversion results at each iteration.

# Apply and update sensitivity weighting as the model updates

# Defining a starting value for the trade-off parameter (beta) between the data
# misfit and the regularization.
starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e1)

# Set the rate of reduction in trade-off parameter (beta) each time the
# the inverse problem is solved. And set the number of Gauss-Newton iterations
# for each trade-off paramter value.
beta_schedule = directives.BetaSchedule(coolingFactor=2.5, coolingRate=2)

# Options for outputting recovered models and predicted data for each beta.
save_iteration = directives.SaveOutputEveryIteration(save_txt=False)

# Setting a stopping criteria for the inversion.
target_misfit = directives.TargetMisfit(chifact=1)

# Apply and update preconditioner as the model updates
update_jacobi = directives.UpdatePreconditioner()

directives_list = [
update_sensitivity_weighting,
starting_beta,
beta_schedule,
save_iteration,
target_misfit,
update_jacobi,
]


## Running the DC Inversion#

To define the inversion object, we need to define the inversion problem and the set of directives. We can then run the inversion.

# Here we combine the inverse problem and the set of directives
dc_inversion = inversion.BaseInversion(
dc_inverse_problem, directiveList=directives_list
)

# Run inversion
recovered_conductivity_model = dc_inversion.run(starting_conductivity_model)

                        SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using same Solver, and solver_opts as the Simulation3DNodal problem***

model has any nan: 0
============================ Inexact Gauss Newton ============================
#     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0  2.55e-02  2.15e+04  0.00e+00  2.15e+04    2.41e+03      0
1  2.55e-02  4.81e+03  1.55e+05  8.77e+03    1.12e+02      0
2  1.02e-02  4.92e+03  1.48e+05  6.43e+03    4.30e+02      0
3  1.02e-02  2.30e+03  3.17e+05  5.54e+03    6.13e+01      0
4  4.08e-03  2.56e+03  2.89e+05  3.74e+03    2.64e+02      0
5  4.08e-03  1.04e+03  5.26e+05  3.18e+03    3.22e+01      0
6  1.63e-03  1.14e+03  4.98e+05  1.95e+03    1.47e+02      0
7  1.63e-03  4.19e+02  7.70e+05  1.68e+03    1.57e+01      0
8  6.53e-04  4.51e+02  7.46e+05  9.38e+02    7.73e+01      0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 2.1469e+03
1 : |xc-x_last| = 1.9103e+01 <= tolX*(1+|x0|) = 8.6952e+01
0 : |proj(x-g)-x|    = 7.7339e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 7.7339e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      15    <= iter          =      9
------------------------- DONE! -------------------------


## Recreate True Conductivity Model#

background_value = 1e-2
conductor_value = 1e-1
resistor_value = 1e-3
true_conductivity_model = background_value * np.ones(nC)

ind_conductor = model_builder.getIndicesSphere(
np.r_[-350.0, 0.0, -300.0], 160.0, mesh.cell_centers[ind_active, :]
)
true_conductivity_model[ind_conductor] = conductor_value

ind_resistor = model_builder.getIndicesSphere(
np.r_[350.0, 0.0, -300.0], 160.0, mesh.cell_centers[ind_active, :]
)
true_conductivity_model[ind_resistor] = resistor_value
true_conductivity_model_log10 = np.log10(true_conductivity_model)


## Plotting True and Recovered Conductivity Model#

# Plot True Model
fig = plt.figure(figsize=(10, 4))

plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan)

ax1 = fig.add_axes([0.15, 0.15, 0.67, 0.75])
mesh.plot_slice(
plotting_map * true_conductivity_model_log10,
ax=ax1,
normal="Y",
ind=int(len(mesh.h) / 2),
grid=False,
clim=(true_conductivity_model_log10.min(), true_conductivity_model_log10.max()),
pcolor_opts={"cmap": mpl.cm.viridis},
)
ax1.set_title("True Conductivity Model")
ax1.set_xlabel("x (m)")
ax1.set_ylabel("z (m)")
ax1.set_xlim([-1000, 1000])
ax1.set_ylim([-1000, 0])

ax2 = fig.add_axes([0.84, 0.15, 0.03, 0.75])
norm = mpl.colors.Normalize(
vmin=true_conductivity_model_log10.min(), vmax=true_conductivity_model_log10.max()
)
cbar = mpl.colorbar.ColorbarBase(
ax2, cmap=mpl.cm.viridis, norm=norm, orientation="vertical", format="$10^{%.1f}$"
)

# Plot recovered model
recovered_conductivity_model_log10 = np.log10(np.exp(recovered_conductivity_model))

fig = plt.figure(figsize=(10, 4))

ax1 = fig.add_axes([0.15, 0.15, 0.67, 0.75])
mesh.plot_slice(
plotting_map * recovered_conductivity_model_log10,
ax=ax1,
normal="Y",
ind=int(len(mesh.h) / 2),
grid=False,
clim=(true_conductivity_model_log10.min(), true_conductivity_model_log10.max()),
pcolor_opts={"cmap": mpl.cm.viridis},
)
ax1.set_title("Recovered Conductivity Model")
ax1.set_xlabel("x (m)")
ax1.set_ylabel("z (m)")
ax1.set_xlim([-1000, 1000])
ax1.set_ylim([-1000, 0])

ax2 = fig.add_axes([0.84, 0.15, 0.03, 0.75])
norm = mpl.colors.Normalize(
vmin=true_conductivity_model_log10.min(), vmax=true_conductivity_model_log10.max()
)
cbar = mpl.colorbar.ColorbarBase(
ax2, cmap=mpl.cm.viridis, norm=norm, orientation="vertical", format="$10^{%.1f}$"
)
plt.show()

• • ## Plotting Normalized Data Misfit or Predicted DC Data#

To see how well the recovered model reproduces the observed data, it is a good idea to compare the predicted and observed data. Here, we accomplish this by plotting the normalized misfit.

# Predicted data from recovered model
dpred_dc = dc_inverse_problem.dpred

# Compute the normalized data misfit
dc_normalized_misfit = (dc_data.dobs - dpred_dc) / dc_data.standard_deviation

if has_plotly:
# Plot IP Data
fig = plot_3d_pseudosection(
dc_data.survey,
dc_normalized_misfit,
scale="linear",
units="",
vlim=[-2, 2],
plane_distance=15,
)

fig.update_layout(
title_text="Normalized Data Misfit",
title_x=0.5,
title_font_size=24,
width=650,
height=500,
scene_camera=dict(
center=dict(x=0, y=0, z=-0.4), eye=dict(x=1.5, y=-1.5, z=1.8)
),
)

plotly.io.show(fig)

else:
print("INSTALL 'PLOTLY' TO VISUALIZE 3D PSEUDOSECTIONS")


## Starting/Reference Model for IP Inversion#

Here, we would create starting and/or reference models for the IP inversion as well as the mapping from the model space to the active cells. Starting and reference models can be a constant background value or contain a-priori structures. Here, the starting model is the 1e-6 V/V.

# Define chargeability model in V/V
air_chargeability = 0.0
background_chargeability = 1e-6

active_map = maps.InjectActiveCells(mesh, ind_active, air_chargeability)
nC = int(ind_active.sum())

chargeability_map = active_map

# Define starting model
starting_chargeability_model = background_chargeability * np.ones(nC)


## Define the Physics of the IP Simulation#

Here, we define the physics of the IP problem. For the chargeability, we require a mapping from the model space to the entire mesh. For the background conductivity/resistivity, we require the conductivity/resistivity on the entire mesh.

ip_simulation = ip.Simulation3DNodal(
mesh,
survey=ip_survey,
etaMap=chargeability_map,
sigma=conductivity_map * recovered_conductivity_model,
solver=Solver,
storeJ=True,
)


## Define IP Inverse Problem#

Here we define the inverse problem in the same manner as the DC inverse problem.

# Define the data misfit (Here we use weighted L2-norm)
ip_data_misfit = data_misfit.L2DataMisfit(data=ip_data, simulation=ip_simulation)

# Define the regularization (model objective function)
ip_regularization = regularization.WeightedLeastSquares(
mesh,
indActive=ind_active,
mapping=maps.IdentityMap(nP=nC),
alpha_s=0.01,
alpha_x=1,
alpha_y=1,
alpha_z=1,
)

# Define how the optimization problem is solved.
ip_optimization = optimization.ProjectedGNCG(
maxIter=15, lower=0.0, upper=10, maxIterCG=30, tolCG=1e-2
)

# Here we define the inverse problem that is to be solved
ip_inverse_problem = inverse_problem.BaseInvProblem(
ip_data_misfit, ip_regularization, ip_optimization
)


## Define IP Inversion Directives#

Here we define the directives in the same manner as the DC inverse problem.

update_sensitivity_weighting = directives.UpdateSensitivityWeights(threshold=1e-3)
starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e2)
beta_schedule = directives.BetaSchedule(coolingFactor=2.5, coolingRate=1)
save_iteration = directives.SaveOutputEveryIteration(save_txt=False)
target_misfit = directives.TargetMisfit(chifact=1.0)
update_jacobi = directives.UpdatePreconditioner()

directives_list = [
update_sensitivity_weighting,
starting_beta,
beta_schedule,
save_iteration,
target_misfit,
update_jacobi,
]

/home/vsts/work/1/s/SimPEG/directives/directives.py:2526: UserWarning:

'threshold' property is deprecated and will be removed in SimPEG 0.20.0.Please use 'threshold_value'.


## Running the IP Inversion#

# Here we combine the inverse problem and the set of directives
ip_inversion = inversion.BaseInversion(
ip_inverse_problem, directiveList=directives_list
)

# Run inversion
recovered_chargeability_model = ip_inversion.run(starting_chargeability_model)

SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.
SimPEG.InvProblem will set Regularization.reference_model to m0.

SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using same Solver, and solver_opts as the Simulation3DNodal problem***

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  2.15e+01  3.73e+03  0.00e+00  3.73e+03    7.88e+02      0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 3.7352e+02
0 : |xc-x_last| = 1.4643e+00 <= tolX*(1+|x0|) = 1.0002e-01
0 : |proj(x-g)-x|    = 7.8764e+02 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 7.8764e+02 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      15    <= iter          =      1
------------------------- DONE! -------------------------


## Recreate True Chargeability Model#

background_value = 1e-6
chargeable_value = 1e-1

true_chargeability_model = background_value * np.ones(nC)
ind_chargeable = model_builder.getIndicesSphere(
np.r_[-350.0, 0.0, -300.0], 160.0, mesh.cell_centers[ind_active, :]
)
true_chargeability_model[ind_chargeable] = chargeable_value


## Plot True and Recovered Chargeability Model#

# Plot True Model
fig = plt.figure(figsize=(10, 4))

plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan)

ax1 = fig.add_axes([0.15, 0.15, 0.67, 0.75])
mesh.plot_slice(
plotting_map * true_chargeability_model,
ax=ax1,
normal="Y",
ind=int(len(mesh.h) / 2),
grid=False,
clim=(true_chargeability_model.min(), true_chargeability_model.max()),
pcolor_opts={"cmap": mpl.cm.plasma},
)
ax1.set_title("True Chargeability Model")
ax1.set_xlabel("x (m)")
ax1.set_ylabel("z (m)")
ax1.set_xlim([-1000, 1000])
ax1.set_ylim([-1000, 0])

ax2 = fig.add_axes([0.84, 0.15, 0.03, 0.75])
norm = mpl.colors.Normalize(
vmin=true_chargeability_model.min(), vmax=true_chargeability_model.max()
)
cbar = mpl.colorbar.ColorbarBase(
ax2, cmap=mpl.cm.plasma, norm=norm, orientation="vertical", format="%.2f"
)
cbar.set_label("Intrinsic Chargeability [V/V]", rotation=270, labelpad=15, size=12)

# Plot Recovered Model
fig = plt.figure(figsize=(10, 4))

ax1 = fig.add_axes([0.15, 0.15, 0.67, 0.75])
mesh.plot_slice(
plotting_map * recovered_chargeability_model,
ax=ax1,
normal="Y",
ind=int(len(mesh.h) / 2),
grid=False,
clim=(true_chargeability_model.min(), true_chargeability_model.max()),
pcolor_opts={"cmap": mpl.cm.plasma},
)
ax1.set_title("Recovered Chargeability Model")
ax1.set_xlabel("x (m)")
ax1.set_ylabel("z (m)")
ax1.set_xlim([-1000, 1000])
ax1.set_ylim([-1000, 0])

ax2 = fig.add_axes([0.84, 0.15, 0.03, 0.75])
norm = mpl.colors.Normalize(
vmin=true_chargeability_model.min(), vmax=true_chargeability_model.max()
)
cbar = mpl.colorbar.ColorbarBase(
ax2, cmap=mpl.cm.plasma, norm=norm, orientation="vertical", format="%.2f"
)
cbar.set_label("Intrinsic Chargeability [V/V]", rotation=270, labelpad=15, size=12)

plt.show()

• • ## Plotting Normalized Data Misfit or Predicted IP Data#

# Predicted data from recovered model
dpred_ip = ip_inverse_problem.dpred

# Normalized misfit
ip_normalized_misfit = (ip_data.dobs - dpred_ip) / ip_data.standard_deviation

if has_plotly:
fig = plot_3d_pseudosection(
ip_data.survey,
ip_normalized_misfit,
scale="linear",
units="",
vlim=[-2, 2],
plane_distance=15,
marker_opts={"colorscale": "plasma"},
)

fig.update_layout(
title_text="Normalized Data Misfit",
title_x=0.5,
title_font_size=24,
width=650,
height=500,
scene_camera=dict(
center=dict(x=0, y=0, z=-0.4), eye=dict(x=1.5, y=-1.5, z=1.8)
),
)

plotly.io.show(fig)

else:
print("INSTALL 'PLOTLY' TO VISUALIZE 3D PSEUDOSECTIONS")


Total running time of the script: ( 4 minutes 27.658 seconds)

Estimated memory usage: 321 MB

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