"""
3D Forward Simulation with User-Defined Waveforms
=================================================
Here we use the module *simpeg.electromagnetics.time_domain* to predict the
TDEM response for a trapezoidal waveform. We consider an airborne survey
which uses a horizontal coplanar geometry. For this tutorial, we focus
on the following:
- How to define the transmitters and receivers
- How to define more complicated transmitter waveforms
- How to define the time-stepping
- How to define the survey
- How to solve TDEM problems on an OcTree mesh
- How to include topography
- The units of the conductivity model and resulting data
Please note that we have used a coarse mesh and larger time-stepping to shorten
the time of the simulation. Proper discretization in space and time is required
to simulate the fields at each time channel with sufficient accuracy.
"""
#########################################################################
# Import Modules
# --------------
#
from discretize import TreeMesh
from discretize.utils import mkvc, refine_tree_xyz, active_from_xyz
from simpeg.utils import plot2Ddata
from simpeg import maps
import simpeg.electromagnetics.time_domain as tdem
from tqdm import tqdm
from simpeg.electromagnetics.time_domain import sources, receivers
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
import os
save_file = False
# sphinx_gallery_thumbnail_number = 3
###############################################################
# Defining Topography
# -------------------
#
# Here we define surface topography as an (N, 3) numpy array. Topography could
# also be loaded from a file. Here we define flat topography, however more
# complex topographies can be considered.
#
xx, yy = np.meshgrid(np.linspace(-3000, 3000, 101), np.linspace(-3000, 3000, 101))
zz = np.zeros(np.shape(xx))
topo_xyz = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)]
###############################################################
# Defining the Waveform
# ---------------------
#
# Under *simpeg.electromagnetic.time_domain.sources*
# there are a multitude of waveforms that can be defined (VTEM, Ramp-off etc...).
# Here, we consider a trapezoidal waveform, which consists of a
# linear ramp-on followed by a linear ramp-off. For each waveform, it
# is important you are cognizant of the off time!!!
#
# Define a discrete set of times for which your transmitter is 'on'. Here
# the waveform is on from -0.002 s to 0 s.
waveform_times = np.linspace(-0.002, 0, 21)
# For each waveform type, you must define the necessary set of kwargs.
# For the trapezoidal waveform we define the ramp on interval, the
# ramp-off interval and the off-time.
waveform = tdem.sources.TrapezoidWaveform(
ramp_on=np.r_[-0.002, -0.001], ramp_off=np.r_[-0.001, 0.0], off_time=0.0
)
# Uncomment to try a quarter sine wave ramp on, followed by a linear ramp-off.
# waveform = tdem.sources.QuarterSineRampOnWaveform(
# ramp_on=np.r_[-0.002, -0.001], ramp_off=np.r_[-0.001, 0.], off_time=0.
# )
# Uncomment to try a custom waveform (just a linear ramp-off). This requires
# defining a function for your waveform.
# def wave_function(t):
# return - t/(np.max(waveform_times) - np.min(waveform_times))
#
# waveform = tdem.sources.RawWaveform(waveform_function=wave_function, off_time=0.)
# Evaluate the waveform for each on time.
waveform_value = [waveform.eval(t) for t in waveform_times]
# Plot the waveform
fig = plt.figure(figsize=(10, 4))
ax1 = fig.add_subplot(111)
ax1.plot(waveform_times, waveform_value, lw=2)
ax1.set_xlabel("Times [s]")
ax1.set_ylabel("Waveform value")
ax1.set_title("Waveform")
#####################################################################
# Create Airborne Survey
# ----------------------
#
# Here we define the survey used in our simulation. For time domain
# simulations, we must define the geometry of the source and its waveform.
# the receivers, we define their geometry, the type of field they measure and
# the time channels at which they measure the field. For this example,
# the survey consists of a uniform grid of airborne measurements.
#
# Observation times for response (time channels)
n_times = 10
time_channels = np.logspace(-5, -2.3, n_times)
# Defining transmitter locations
n_tx =4
xtx, ytx, ztx = np.meshgrid(
np.linspace(-50, 50, n_tx), np.linspace(-50, 50, n_tx), [40]
)
source_locations = np.c_[mkvc(xtx), mkvc(ytx), mkvc(ztx)]
ntx = np.size(xtx)
# Define receiver locations
xrx, yrx, zrx = np.meshgrid(
np.linspace(-50, 50, n_tx), np.linspace(-50, 50, n_tx), [30]
)
receiver_locations = np.c_[mkvc(xrx), mkvc(yrx), mkvc(zrx)]
source_list = [] # Create empty list to store sources
# Each unique location defines a new transmitter
for ii in range(ntx):
# Here we define receivers that measure the h-field in A/m
dbzdt_receiver = tdem.receivers.PointMagneticFluxTimeDerivative(
receiver_locations[ii, :], time_channels, "z"
)
receivers_list = [
dbzdt_receiver
] # Make a list containing all receivers even if just one
# Must define the transmitter properties and associated receivers
source = sources.CircularLoop(
receivers_list,
location=source_locations[ii], # 线圈中心位置
waveform=waveform, # 使用前面定义的波形
radius=5.0, # 设置线圈半径（单位：m）
orientation="z", # 线圈法向方向
current=50.0 # 发射电流（A）
)
source_list.append(source)
survey = tdem.Survey(source_list)
###############################################################
# Create OcTree Mesh
# ------------------
#
# Here we define the OcTree mesh that is used for this example.
# We chose to design a coarser mesh to decrease the run time.
# When designing a mesh to solve practical time domain problems:
#
# - Your smallest cell size should be 10%-20% the size of your smallest diffusion distance
# - The thickness of your padding needs to be 2-3 times biggest than your largest diffusion distance
# - The diffusion distance is ~1260*np.sqrt(rho*t)
#
#
dh = 2.0 # base cell width
dom_width = 400.0 # domain width
nbc = 2 ** int(np.round(np.log(dom_width / dh) / np.log(2.0))) # num. base cells
# Define the base mesh
h = [(dh, nbc)]
mesh = TreeMesh([h, h, h], x0="CCC")
# Mesh refinement based on topography
mesh = refine_tree_xyz(
mesh, topo_xyz, octree_levels=[0, 0, 1], method="surface", finalize=False
)
# Mesh refinement near transmitters and receivers
mesh = refine_tree_xyz(
mesh, receiver_locations, octree_levels=[ 2,4], method="radial", finalize=False
)
# Refine core mesh region
xp, yp, zp = np.meshgrid([-80.0, 80.0], [-80.0, 80.0], [-60.0, 0.0])
xyz = np.c_[mkvc(xp), mkvc(yp), mkvc(zp)]
mesh = refine_tree_xyz(mesh, xyz, octree_levels=[0, 2,4], method="box", finalize=False)
mesh.finalize()
###############################################################
# Create Conductivity Model and Mapping for OcTree Mesh
# -----------------------------------------------------
#
# Here, we define the electrical properties of the Earth as a conductivity
# model. The model consists of a conductive block within a more
# resistive background.
#
# Conductivity in S/m
air_conductivity = 1e-8
background_conductivity = 2e-3
block_conductivity = 2e0
# Active cells are cells below the surface.
ind_active = active_from_xyz(mesh, topo_xyz)
if ind_active.sum() == 0:
raise ValueError("No active cells detected below topography — check topo or mesh alignment.")
model_map = maps.InjectActiveCells(mesh, ind_active, air_conductivity)
# Define the model
model = background_conductivity * np.ones(ind_active.sum())
ind_block = (
(mesh.gridCC[ind_active, 0] < 10.0)
& (mesh.gridCC[ind_active, 0] > -10.0)
& (mesh.gridCC[ind_active, 1] < 10.0)
& (mesh.gridCC[ind_active, 1] > -10.0)
& (mesh.gridCC[ind_active, 2] > -20.0)
& (mesh.gridCC[ind_active, 2] < -15.0)
)
model[ind_block] = block_conductivity
# Plot log-conductivity model
mpl.rcParams.update({"font.size": 12})
fig = plt.figure(figsize=(7, 6))
log_model = np.log10(model)
plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan)
ax1 = fig.add_axes([0.13, 0.1, 0.6, 0.85])
mesh.plot_slice(
plotting_map * log_model,
normal="Y",
ax=ax1,
ind=int(mesh.h[0].size / 2),
grid=True,
clim=(np.min(log_model), np.max(log_model)),
)
ax1.set_title("Conductivity Model at Y = 0 m")
ax2 = fig.add_axes([0.75, 0.1, 0.05, 0.85])
norm = mpl.colors.Normalize(vmin=np.min(log_model), vmax=np.max(log_model))
cbar = mpl.colorbar.ColorbarBase(
ax2, norm=norm, orientation="vertical", format="$10^{%.1f}$"
)
cbar.set_label("Conductivity [S/m]", rotation=270, labelpad=15, size=12)
######################################################
# Define the Time-Stepping
# ------------------------
#
# Stuff about time-stepping and some rule of thumb
#
time_steps = [(5e-6, 10), (1e-5, 10), (5e-5, 10), (1e-4, 20), (5e-4, 10), (1e-3, 5)
]
#######################################################################
# Simulation: Time-Domain Response
# --------------------------------
#
# Here we define the formulation for solving Maxwell's equations. Since we are
# measuring the time-derivative of the magnetic flux density and working with
# a resistivity model, the EB formulation is the most natural. We must also
# remember to define the mapping for the conductivity model.
# We defined a waveform 'on-time' is from -0.002 s to 0 s. As a result, we need
# to set the start time for the simulation to be at -0.002 s.
#
simulation = tdem.simulation.Simulation3DMagneticFluxDensity(
mesh, survey=survey, sigmaMap=model_map, t0=-0.002
)
# Set the time-stepping for the simulation
simulation.time_steps = time_steps
########################################################################
# Predict Data and Plot
# ---------------------
print("开始进行 3D 正演模拟 (dpred)...")
# SimPEG 的 dpred 实际上是计算每个源的响应，然后拼接起来
# 我们可以手动循环源列表，并用 tqdm 跟踪进度
dpred_list = []
# 使用 tqdm 包装 simulation.survey.source_list 来显示进度
for src in tqdm(simulation.survey.source_list, desc="计算每个发射源的响应"):
# 临时创建一个只包含当前源的 Survey 对象
temp_survey = tdem.Survey([src])
# 将 simulation 对象的 survey 临时设置为 temp_survey
original_survey = simulation.survey
simulation.survey = temp_survey
# 预测当前源的数据
# 注意: SimPEG 模型的 dpred 方法需要完整的模型，即使只计算一个源
# 如果 SimPEG 版本较新，且支持对单个源进行计算，可能更简洁
# 否则，这种临时替换 survey 的方式是通用的
dpred_list.append(simulation.dpred(model))
# 恢复 simulation 对象的 survey
simulation.survey = original_survey
# 将所有源的预测数据拼接起来
dpred = np.concatenate(dpred_list)
print("3D 正演模拟完成。")
# Predict data for a given model
dpred = simulation.dpred(model)
# Data were organized by location, then by time channel
dpred_plotting = np.reshape(dpred, (n_tx**2, n_times))
# Plot
fig = plt.figure(figsize=(10, 4))
# dB/dt at early time
v_max = np.max(np.abs(dpred_plotting[:, 0]))
ax11 = fig.add_axes([0.05, 0.05, 0.35, 0.9])
plot2Ddata(
receiver_locations[:, 0:2],
dpred_plotting[:, 0],
ax=ax11,
ncontour=30,
clim=(-v_max, v_max),
contourOpts={"cmap": "bwr"},
)
ax11.set_title("dBz/dt at 0.0001 s")
ax12 = fig.add_axes([0.42, 0.05, 0.02, 0.9])
norm1 = mpl.colors.Normalize(vmin=-v_max, vmax=v_max)
cbar1 = mpl.colorbar.ColorbarBase(
ax12, norm=norm1, orientation="vertical", cmap=mpl.cm.bwr
)
cbar1.set_label("$T/s$", rotation=270, labelpad=15, size=12)
# dB/dt at later times
v_max = np.max(np.abs(dpred_plotting[:, -1]))
ax21 = fig.add_axes([0.55, 0.05, 0.35, 0.9])
plot2Ddata(
receiver_locations[:, 0:2],
dpred_plotting[:, -1],
ax=ax21,
ncontour=30,
clim=(-v_max, v_max),
contourOpts={"cmap": "bwr"},
)
ax21.set_title("dBz/dt at 0.001 s")
ax22 = fig.add_axes([0.92, 0.05, 0.02, 0.9])
norm2 = mpl.colors.Normalize(vmin=-v_max, vmax=v_max)
cbar2 = mpl.colorbar.ColorbarBase(
ax22, norm=norm2, orientation="vertical", cmap=mpl.cm.bwr
)
cbar2.set_label("$T/s$", rotation=270, labelpad=15, size=12)
plt.show()
#######################################################
# Optional: Export Data
# ---------------------
for i in range(n_times):
fig = plt.figure(figsize=(6, 4))
v_max = np.max(np.abs(dpred_plotting[:, i]))
ax = fig.add_axes([0.1, 0.1, 0.75, 0.8])
plot2Ddata(
receiver_locations[:, 0:2],
dpred_plotting[:, i],
ax=ax,
ncontour=30,
clim=(-v_max, v_max),
contourOpts={"cmap": "bwr"},
)
ax.set_title(f"dBz/dt at t = {time_channels[i]:.6f} s")
# 加色条
cax = fig.add_axes([0.88, 0.1, 0.03, 0.8])
norm = mpl.colors.Normalize(vmin=-v_max, vmax=v_max)
cbar = mpl.colorbar.ColorbarBase(cax, norm=norm, orientation="vertical", cmap=mpl.cm.bwr)
cbar.set_label("$T/s$", rotation=270, labelpad=15, size=12)
plt.show()
# Write the true model, data and topography
#
if save_file:
dir_path = os.path.dirname(tdem.__file__).split(os.path.sep)[:-3]
dir_path.extend(["tutorials", "assets", "tdem"])
dir_path = os.path.sep.join(dir_path) + os.path.sep
fname = dir_path + "tdem_topo.txt"
np.savetxt(fname, np.c_[topo_xyz], fmt="%.4e")
# Write data with 2% noise added
fname = dir_path + "tdem_data.obs"
dpred = dpred + 0.02 * np.abs(dpred) * np.random.randn(len(dpred))
t_vec = np.kron(np.ones(ntx), time_channels)
receiver_locations = np.kron(receiver_locations, np.ones((len(time_channels), 1)))
np.savetxt(fname, np.c_[receiver_locations, t_vec, dpred], fmt="%.4e")
# Plot true model
output_model = plotting_map * model
output_model[np.isnan(output_model)] = 1e-8
fname = dir_path + "true_model.txt"
np.savetxt(fname, output_model, fmt="%.4e")
此代码中，正演模拟time_channels和time_steps如何设置参数
最新发布
本文提供两个链接资源，分别介绍视觉定位系统的估计方法及数学在线课程。前者由INRIA研究机构提供，深入探讨了机器人视觉定位中目标估计的技术细节；后者则是一门在线数学课程，覆盖了与视觉定位密切相关的数学基础知识。
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