"""
A simulation is the computation (modelling) of the electromagnetic responses
due to a given model and survey.
The simulation module combines the different pieces of ``emg3d`` providing
a high-level, specialised modelling tool for the end user.
"""
# Copyright 2018-2021 The emsig community.
#
# This file is part of emg3d.
#
# Licensed under the Apache License, Version 2.0 (the "License"); you may not
# use this file except in compliance with the License. You may obtain a copy
# of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
# WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
# License for the specific language governing permissions and limitations under
# the License.
import os
import warnings
import itertools
from pathlib import Path
from copy import deepcopy
import numpy as np
from emg3d import fields, io, maps, meshes, models, solver, surveys, utils
__all__ = ['Simulation', ]
[docs]@utils._known_class
class Simulation:
"""Create a simulation for a given survey on a given model.
A simulation can be used to compute responses for an entire survey, hence
for an arbitrary amount of sources, receivers, and frequencies. The
responses are computed in parallel over sources and frequencies. It can
also be used to compute the misfit with the data and to compute the
gradient of the misfit function.
The computational grid(s) can either be provided, or automatic gridding can
be used; see the description of the parameters ``gridding`` and
``gridding_opts`` for more details.
.. note::
The automatic gridding does its best to generate meshes that are
suitable for the provided model and survey. However, CSEM spans a wide
range of acquisition layouts, and both frequencies and conductivities
or resistivities span many orders of magnitude. This makes it hard to
have a function that fits all purposes. Check the meshes with your
expert knowledge. Also, the automatic gridding is conservative in its
estimate, trying to be on the save side (correct results over speed).
This means, however, that often smaller grids could be used by
providing the appropriate options in ``gridding_opts`` or directly
providing your own computational grids.
.. note::
The package ``xarray`` has to be installed in order to use
``Simulation``:
``pip install xarray`` or ``conda install -c conda-forge xarray``.
Parameters
----------
survey : Survey
The survey; a :class:`emg3d.surveys.Survey` instance. The survey
contains sources, receivers, frequencies, and optionally data.
The survey-data will be modified in place. Provide ``survey.copy()`` if
you want to avoid this.
model : Model
The model; a :class:`emg3d.models.Model` instance.
max_workers : int, default: 4
The maximum number of processes that can be used to execute the
given calls.
gridding : str, default: 'single'
Method to create the computational grids.
The different methods are:
- ``'same'``: Same grid as for the input model.
- ``'single'``: A single grid for all sources and frequencies.
- ``'frequency'``: Frequency-dependent grids.
- ``'source'``: Source-dependent grids.
- ``'both'``: Frequency- and source-dependent grids.
- ``'input'``: Same as ``'single'``, but the grid has to be provided in
``gridding_opts`` instead of being automatically created.
- ``'dict'``: Same as ``'both'``, but the grids have to be provided in
``gridding_opts`` in the form of ``dict[source][frequency]`` instead
of being automatically created.
See the parameter ``gridding_opts`` for more details.
If ``gridding`` is ``'same'`` or ``'input'``, the input grid is checked
if it is a sensible grid for emg3d; if not, it throws a warning. In the
other cases the grids are created by emg3d itself, they will be fine.
(If ``'dict'`` we assume the user knows how to provide good grids.)
gridding_opts : {dict, TensorMesh}, default: {}
Input format depends on ``gridding``:
- ``'same'``: Nothing, ``gridding_opts`` is not permitted.
- ``'single'``, ``'frequency'``, ``'source'``, ``'both'``: Described
below.
- ``'input'``: A :class:`emg3d.meshes.TensorMesh`.
- ``'dict'``: Dictionary of the format ``dict[source][frequency]``
containing a :class:`emg3d.meshes.TensorMesh` for each
source-frequency pair.
The dict in the case of ``'single'``, ``'frequency'``, ``'source'``,
``'both``' is passed to :func:`emg3d.meshes.construct_mesh`; consult
the corresponding documentation for more information. Parameters that
are not provided are estimated from the provided model, grid, and
survey using :func:`emg3d.meshes.estimate_gridding_opts`, which
documentation contains more information too.
There are two notably differences to the parameters described in
:func:`emg3d.meshes.construct_mesh`:
- ``vector``: besides the normal possibility it can also be a string
containing one or several of ``'x'``, ``'y'``, and ``'z'``. In these
cases the corresponding dimension of the input mesh is provided as
vector. See :func:`emg3d.meshes.estimate_gridding_opts`.
- ``expand``: in the format of ``[property_sea, property_air]``; if
provided, the input model is expanded up to the seasurface with sea
water, and an air layer is added. The actual height of the seasurface
can be defined with the key ``seasurface``. See
:func:`emg3d.models.expand_grid_model`.
solver_opts : dict, default: {'verb': 1'}
Passed through to :func:`emg3d.solver.solve`. The dict can contain any
parameter that is accepted by the :func:`emg3d.solver.solve` except for
``model``, ``sfield``, ``efield``, ``return_info``, and ``log``.
Default verbosity is ``verb=2``.
verb : int, default: 0
Level of verbosity. Possible options:
- -1: Errors.
- 0: Warnings.
- 1: Info.
name : str, default: None
Name of the simulation.
info : str, default: None
Simulation info or any other info (e.g., what was the purpose of this
simulation).
receiver_interpolation : str, default: 'cubic':
Interpolation method to obtain the response at receiver location;
'cubic' or 'linear'. Cubic is more precise. However, if you are
interested in the gradient, you need to choose 'linear' at the moment,
as there are only linearly interpolated source functions. To be the
proper adjoint for the gradient the receiver has to be interpolated
linearly too.
file_dir : str, default: None
Absolute or relative path, where temporary files should be stored. By
default, everything is done in memory. However, for large models with
many sources and frequencies this can become memory consuming.
Providing a ``file_dir`` can help with this. Fields and models are then
stored to disk, and each process accesses the files it needs. There is
only a gain if there are more source-frequency pairs than concurrent
running processes.
Note that the directory is created if it does not exist. However, the
parent directory must exist.
Also note that the files are stored as .h5-files, and you need to have
``h5py`` installed to use this feature.
"""
# Gridding descriptions (for repr's).
_gridding_descr = {
'same': 'Same grid as for model',
'single': 'Single grid for all sources and frequencies',
'frequency': 'Frequency-dependent grids',
'source': 'Source-dependent grids',
'both': 'Frequency- and source-dependent grids',
'input': 'Provided grid, same for all sources/frequencies',
'dict': 'Provided grids, frequency-/source-dependent',
}
def __init__(self, survey, model, max_workers=4, gridding='single',
**kwargs):
"""Initiate a new Simulation instance."""
# Store some inputs as is, optional ones with defaults.
self.survey = survey
self.max_workers = max_workers
self.gridding = gridding
self.verb = kwargs.pop('verb', 0)
self.name = kwargs.pop('name', None)
self.info = kwargs.pop('info', None)
self.receiver_interpolation = kwargs.pop(
'receiver_interpolation', 'cubic')
# Assemble solver_opts.
self.solver_opts = {
'verb': 1, # Default verbosity, can be overwritten.
'log': -1, # Default only log, can be overwritten.
**kwargs.pop('solver_opts', {}), # User setting.
'return_info': True, # return_info=True is forced.
}
# Initiate dictionaries and other values with None's.
self._dict_grid = self._dict_initiate
self._dict_efield = self._dict_initiate
self._dict_efield_info = self._dict_initiate
self._gradient = None
self._misfit = None
# Initiate file_dir
self.file_dir = kwargs.pop('file_dir', None)
if self.file_dir:
self.file_dir = os.path.abspath(self.file_dir)
# Create directory if it doesn't exist yet.
Path(self.file_dir).mkdir(exist_ok=True)
# Get model taking gridding_opts into account.
# Sets self.model and self.gridding_opts.
self._set_model(model, kwargs)
# Initiate synthetic data with NaN's if they don't exist.
if 'synthetic' not in self.survey.data.keys():
self.survey._data['synthetic'] = self.data.observed.copy(
data=np.full(self.survey.shape, np.nan+1j*np.nan))
# `tqdm`-options; undocumented.
# Can be used to, e.g., disable tqdm completely via:
# > emg3d.Simulation(args, kwargs, tqdm_opts={'disable': True})
tqdm_opts = kwargs.pop('tqdm_opts', {})
self._tqdm_opts = {
**tqdm_opts,
**{'bar_format': '{desc}: {bar}{n_fmt}/{total_fmt} [{elapsed}]'},
}
# Ensure no kwargs left.
if kwargs:
raise TypeError(f"Unexpected **kwargs: {list(kwargs.keys())}.")
# Check the grid if one was explicitly provided.
if gridding == 'same':
meshes.check_mesh(self.model.grid)
elif gridding == 'input':
meshes.check_mesh(self._grid_single)
def __repr__(self):
"""Simple representation."""
name = f" «{self.name}»" if self.name else ""
info = f"{self.info}\n" if self.info else ""
survey = f" «{self.survey.name}»" if self.survey.name else ""
return (f":: {self.__class__.__name__}{name} ::\n{info}\n"
f"- {self.survey.__class__.__name__}{survey}: "
f"{self.survey.shape[0]} sources; "
f"{self.survey.shape[1]} receivers; "
f"{self.survey.shape[2]} frequencies\n"
f"- {self.model.__repr__()}\n"
f"- Gridding: {self._gridding_descr[self.gridding]}; "
f"{self._info_grids}")
def _repr_html_(self):
"""HTML representation."""
name = f" «{self.name}»" if self.name else ""
info = f"{self.info}<br>" if self.info else ""
survey = f" «{self.survey.name}»" if self.survey.name else ""
return (f"<h3>{self.__class__.__name__}{name}</h3>{info}"
f"<ul>"
f" <li>{self.survey.__class__.__name__}{survey}:"
f" {self.survey.shape[0]} sources;"
f" {self.survey.shape[1]} receivers;"
f" {self.survey.shape[2]} frequencies</li>"
f" <li>{self.model.__repr__()}</li>"
f" <li>Gridding: {self._gridding_descr[self.gridding]}; "
f" {self._info_grids}</li>"
f"</ul>")
[docs] def clean(self, what='computed'):
"""Clean part of the data base.
Parameters
----------
what : str, default: 'computed'
What to clean. Possibilities:
- ``'computed'``:
Removes all computed properties: electric and magnetic fields and
responses at receiver locations.
- ``'keepresults'``:
Removes everything except for the responses at receiver
locations.
- ``'all'``:
Removes everything (leaves it plain as initiated).
"""
if what not in ['computed', 'keepresults', 'all']:
raise TypeError(f"Unrecognized `what`: {what}.")
# Clean grid/model-dicts.
if what in ['keepresults', 'all']:
# These exist always and have to be initiated.
for name in ['_dict_grid', ]:
delattr(self, name)
setattr(self, name, self._dict_initiate)
# Clean field-dicts.
if what in ['computed', 'keepresults', 'all']:
# These exist always and have to be initiated.
for name in ['_dict_efield', '_dict_efield_info']:
delattr(self, name)
setattr(self, name, self._dict_initiate)
# These only exist with gradient; don't initiate them.
for name in ['_dict_bfield', '_dict_bfield_info']:
if hasattr(self, name):
delattr(self, name)
# Remove files if they exist.
if self.file_dir:
for p in Path(self.file_dir).glob('[ebg]field_*.h5'):
p.unlink()
# Clean data.
if what in ['computed', 'all']:
for key in ['residual', 'weight']:
if key in self.data.keys():
del self.data[key]
self.data['synthetic'] = self.data.observed.copy(
data=np.full(self.survey.shape, np.nan+1j*np.nan))
for name in ['_gradient', '_misfit']:
delattr(self, name)
setattr(self, name, None)
[docs] def copy(self, what='computed'):
"""Return a copy of the Simulation.
See ``to_file`` for more information regarding ``what``.
"""
return self.from_dict(self.to_dict(what, True))
[docs] def to_dict(self, what='computed', copy=False):
"""Store the necessary information of the Simulation in a dict.
See `to_file` for more information regarding `what`.
"""
# If to_dict is called from to_file, it has a _what_to_file attribute.
if hasattr(self, '_what_to_file'):
what = self._what_to_file
delattr(self, '_what_to_file')
if what not in ['computed', 'results', 'all', 'plain']:
raise TypeError(f"Unrecognized `what`: {what}.")
# Initiate dict with input parameters.
out = {
'__class__': self.__class__.__name__,
'survey': self.survey.to_dict(),
'model': self.model.to_dict(),
'max_workers': self.max_workers,
'gridding': self.gridding,
'gridding_opts': self.gridding_opts,
'solver_opts': self.solver_opts,
'verb': self.verb,
'name': self.name,
'info': self.info,
'tqdm_opts': self._tqdm_opts,
'receiver_interpolation': self.receiver_interpolation,
'file_dir': self.file_dir,
'_input_sc2': self._input_sc2,
}
# Clean unwanted data if plain.
if what == 'plain':
for key in ['synthetic', 'residual', 'weights']:
if key in out['survey']['data'].keys():
del out['survey']['data'][key]
# Store wanted dicts.
if what in ['computed', 'all']:
for name in ['_dict_efield', '_dict_efield_info',
'_dict_bfield', '_dict_bfield_info']:
if hasattr(self, name):
out[name] = getattr(self, name)
if what == 'all':
for name in ['_dict_grid', ]:
if hasattr(self, name):
out[name] = getattr(self, name)
# Store gradient and misfit.
if what in ['computed', 'results', 'all']:
out['gradient'] = self._gradient
out['misfit'] = self._misfit
if copy:
return deepcopy(out)
else:
return out
[docs] @classmethod
def from_dict(cls, inp):
"""Convert dict into :class:`emg3d.simulations.Simulation` instance.
Parameters
----------
inp : dict
Dictionary as obtained from
:func:`emg3d.simulations.Simulation.to_dict`.
Returns
-------
simulation : Simulation
A :class:`emg3d.simulations.Simulation` instance.
"""
inp = {k: v for k, v in inp.items() if k != '__class__'}
# Get all class-inputs.
inp_names = ['survey', 'model', 'max_workers', 'gridding',
'solver_opts', 'verb', 'name', 'info']
cls_inp = {k: inp.pop(k) for k in inp_names}
cls_inp['gridding_opts'] = inp.pop('gridding_opts', {})
cls_inp['survey'] = surveys.Survey.from_dict(cls_inp['survey'])
cls_inp['model'] = models.Model.from_dict(cls_inp['model'])
input_sc2 = inp.pop('_input_sc2', False)
if input_sc2:
cls_inp['_input_sc2'] = input_sc2
cls_inp['receiver_interpolation'] = inp.pop(
'receiver_interpolation', 'cubic')
cls_inp['file_dir'] = inp.pop('file_dir', None)
cls_inp['tqdm_opts'] = inp.pop('tqdm_opts', {})
# Instantiate the class.
out = cls(**cls_inp)
# Add existing derived/computed properties.
data = ['_dict_grid',
'_dict_efield', '_dict_efield_info',
'_dict_bfield', '_dict_bfield_info']
for name in data:
if name in inp.keys():
values = inp.pop(name)
# De-serialize Model, Field, and TensorMesh instances.
io._dict_deserialize(values)
setattr(out, name, values)
# Add gradient and misfit.
data = ['gradient', 'misfit']
for name in data:
if name in inp.keys():
setattr(out, '_'+name, inp.pop(name))
return out
[docs] def to_file(self, fname, what='computed', name='simulation', **kwargs):
"""Store Simulation to a file.
Parameters
----------
fname : str
Absolute or relative file name including ending, which defines the
used data format. See :func:`emg3d.io.save` for the options.
what : str, default: 'computed'
What to store. Possibilities:
- ``'computed'``:
Stores all computed properties: electric fields and responses at
receiver locations.
- '``results'``:
Stores only the response at receiver locations.
- ``'all'``:
Stores everything. Note that if ``file_dir`` is set, these files
will remain there.
- ``'plain'``:
Only stores the plain Simulation (as initiated).
name : str, default: 'simulation'
Name with which the simulation is stored in the file.
kwargs : Keyword arguments, optional
Passed through to :func:`emg3d.io.save`.
"""
# Add what to self, will be removed in to_dict.
self._what_to_file = what
# Add simulation to dict.
kwargs[name] = self
# If verb is not defined, use verbosity of simulation.
kwargs['verb'] = kwargs.get('verb', self.verb)
return io.save(fname, **kwargs)
[docs] @classmethod
def from_file(cls, fname, name='simulation', **kwargs):
"""Load Simulation from a file.
Parameters
----------
fname : str
Absolute or relative file name including extension.
name : str, default: 'simulation'
Name under which the simulation is stored within the file.
kwargs : Keyword arguments, optional
Passed through to :func:`io.load`.
Returns
-------
simulation : Simulation
A :class:`emg3d.simulations.Simulation` instance.
info : str, returned if verb<0
Info-string.
"""
out = io.load(fname, **kwargs)
if 'verb' in kwargs and kwargs['verb'] < 0:
return out[0][name], out[1]
else:
return out[name]
# GET FUNCTIONS
@property
def data(self):
"""Shortcut to survey.data."""
return self.survey.data
[docs] def get_grid(self, source, frequency):
"""Return computational grid of the given source and frequency."""
freq = self._freq_inp2key(frequency)
# Return grid if it exists already.
if self._dict_grid[source][freq] is not None:
return self._dict_grid[source][freq]
# Same grid as for provided model.
if self.gridding == 'same':
# Store link to grid.
self._dict_grid[source][freq] = self.model.grid
# Frequency-dependent grids.
elif self.gridding == 'frequency':
# Initiate dict.
if not hasattr(self, '_grid_frequency'):
self._grid_frequency = {}
# Get grid for this frequency if not yet computed.
if freq not in self._grid_frequency.keys():
# Get grid and store it.
inp = {**self.gridding_opts, 'frequency':
self.survey.frequencies[freq]}
self._grid_frequency[freq] = meshes.construct_mesh(**inp)
# Store link to grid.
self._dict_grid[source][freq] = self._grid_frequency[freq]
# Source-dependent grids.
elif self.gridding == 'source':
# Initiate dict.
if not hasattr(self, '_grid_source'):
self._grid_source = {}
# Get grid for this source if not yet computed.
if source not in self._grid_source.keys():
# Get grid and store it.
center = self.survey.sources[source].center
inp = {**self.gridding_opts, 'center': center}
self._grid_source[source] = meshes.construct_mesh(**inp)
# Store link to grid.
self._dict_grid[source][freq] = self._grid_source[source]
# Source- and frequency-dependent grids.
elif self.gridding == 'both':
# Get grid and store it.
center = self.survey.sources[source].center
inp = {**self.gridding_opts, 'frequency':
self.survey.frequencies[freq], 'center': center}
self._dict_grid[source][freq] = meshes.construct_mesh(**inp)
# Use a single grid for all sources and receivers.
# Default case; catches 'single' but also anything else.
else:
# Get grid if not yet computed.
if not hasattr(self, '_grid_single'):
# Get grid and store it.
self._grid_single = meshes.construct_mesh(**self.gridding_opts)
# Store link to grid.
self._dict_grid[source][freq] = self._grid_single
# Use recursion to return grid.
return self.get_grid(source, frequency)
[docs] def get_model(self, source, frequency):
"""Return model on the grid of the given source and frequency."""
grid = self.get_grid(source, self._freq_inp2key(frequency))
return self.model.interpolate_to_grid(grid)
[docs] def get_efield(self, source, frequency):
"""Return electric field for given source and frequency."""
freq = self._freq_inp2key(frequency)
# If it doesn't exist yet, compute it.
if self._dict_get('efield', source, freq) is None:
self.compute(source=source, frequency=freq)
return self._dict_get('efield', source, freq)
[docs] def get_hfield(self, source, frequency):
"""Return magnetic field for given source and frequency."""
freq = self._freq_inp2key(frequency)
# If electric field not computed yet compute it.
if self._dict_get('efield', source, freq) is None:
self.compute(source=source, frequency=freq)
# Return magnetic field.
return fields.get_magnetic_field(
self.get_model(source, freq),
self._dict_get('efield', source, freq),
)
[docs] def get_efield_info(self, source, frequency):
"""Return the solver information of the corresponding computation."""
freq = self._freq_inp2key(frequency)
return self._dict_get('efield_info', source, freq)
def _dict_get(self, which, source, frequency):
"""Return source-frequency pair from dictionary `which`.
Thin wrapper for ``self._dict_{which}[{source}][{frequency}]``, that
works as well for file-based computations.
"""
value = getattr(self, f"_dict_{which}")[source][frequency]
return self._load(value, ['efield', 'info']['info' in which])
def _load(self, value, what):
"""Returns `value` (memory) or loads `value[what]` (files)."""
if self.file_dir and value is not None:
return io.load(value, verb=0)[what]
else:
return value
def _data_or_file(self, what, source, frequency, data):
"""Return data or file-name for given what, source, and frequency."""
if self.file_dir:
fname = os.path.join(
self.file_dir, f"{what}_{source}_{frequency}.h5")
io.save(fname, data=data, verb=0)
return fname
else:
return data
def _get_responses(self, source, frequency, efield=None):
"""Return electric and magnetic fields at receiver locations."""
# Get receiver types and their coordinates.
erec, mrec = self.survey._irec_types
erec_coord, mrec_coord = self.survey._rec_types_coord(source)
# Initiate output.
resp = np.zeros_like(self.data.synthetic.loc[source, :, frequency])
# efield of this source/frequency if not provided.
if efield is None:
efield = self._dict_get('efield', source, frequency)
if erec.size:
# Store electric receivers.
resp[erec] = efield.get_receiver(
receiver=erec_coord, method=self.receiver_interpolation,
)
if mrec.size:
# Compute magnetic field.
hfield = fields.get_magnetic_field(
self.get_model(source, frequency), efield,
)
# Store electric receivers.
resp[mrec] = hfield.get_receiver(
receiver=mrec_coord, method=self.receiver_interpolation,
)
return resp
# ASYNCHRONOUS COMPUTATION
[docs] def compute(self, observed=False, **kwargs):
"""Compute efields asynchronously for all sources and frequencies.
Parameters
----------
observed : bool, default: False
If True, it stores the current `synthetic` responses also as
`observed` responses.
add_noise : bool, default: True
Boolean if to add noise to observed data (if ``observed=True``).
All remaining ``kwargs`` are forwarded to
:meth:`emg3d.surveys.Survey.add_noise`.
"""
# Undocumented (internal):
# If the call is from `get_efield`, it will have source/frequency.
# This use is only internal. End users should use `get_efield()`.
srcfreq = [
(kwargs.pop('source', None), kwargs.pop('frequency', None)),
]
if not srcfreq[0][0]:
# "Normal" case: all source-frequency pairs.
srcfreq = self._srcfreq
# Create iterable from src/freq-list to call the process_map.
def collect_efield_inputs(inp):
"""Collect inputs."""
source, freq = inp
data = {
'model': self.model,
'grid': self.get_grid(source, freq),
'source': self.survey.sources[source],
'frequency': self.survey.frequencies[freq],
# efield is None if not comp. yet; else it is the solution.
'efield': self._dict_get('efield', source, freq),
'solver_opts': self.solver_opts,
}
return self._data_or_file('efield', source, freq, data)
# Use process_map to compute fields in parallel.
out = utils._process_map(
solver._solve,
list(map(collect_efield_inputs, self._srcfreq)),
max_workers=self.max_workers,
**{'desc': 'Compute efields', **self._tqdm_opts},
)
# Loop over src-freq combinations to extract and store.
for i, (src, freq) in enumerate(srcfreq):
# Store efield and solver info.
self._dict_efield[src][freq] = out[i][0]
self._dict_efield_info[src][freq] = out[i][1]
# Store responses at receiver locations.
resp = self._get_responses(src, freq)
self.data['synthetic'].loc[src, :, freq] = resp
# Print solver info.
self.print_solver_info('efield', verb=self.verb)
# If it shall be used as observed data save a copy.
if observed:
# Copy synthetic to observed.
self.data['observed'] = self.data['synthetic'].copy()
# Add noise.
if kwargs.get('add_noise', True):
self.survey.add_noise(**kwargs)
# OPTIMIZATION
@property
def gradient(self):
r"""Compute the discrete gradient using the adjoint-state method.
The discrete adjoint-state gradient for a single source at a single
frequency is given by Equation (10) in [PlMu08]_,
.. math::
\nabla_p \phi(\textbf{p}) =
-&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{x; k+\frac{1}{2}, l, m}
\frac{\partial S_{k+\frac{1}{2}, l, m}}{\partial \textbf{p}}
\textbf{E}_{x; k+\frac{1}{2}, l, m}\\
-&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{y; k, l+\frac{1}{2}, m}
\frac{\partial S_{k, l+\frac{1}{2}, m}}{\partial \textbf{p}}
\textbf{E}_{y; k, l+\frac{1}{2}, m}\\
-&\sum_{k,l,m}\mathbf{\bar{\lambda}}_{z; k, l, m+\frac{1}{2}}
\frac{\partial S_{k, l, m+\frac{1}{2}}}{\partial \textbf{p}}
\textbf{E}_{z; k, l, m+\frac{1}{2}}\, ,
where :math:`\textbf{E}` is the electric (forward) field and
:math:`\mathbf{\lambda}` is the back-propagated residual field (from
electric and magnetic receivers); :math:`\bar{~}` denotes conjugate.
The :math:`\partial S`-part takes care of the volume-averaged model
parameters.
.. warning::
To obtain the proper adjoint-state gradient you have to choose
linear interpolation for the receiver responses:
``emg3d.Simulation(..., receiver_interpolation='linear')``. The
reason is that the point-source is the adjoint of a tri-linear
interpolation, so the residual should come from a linear
interpolation.
.. note::
The currently implemented gradient is only for isotropic models
without relative electric permittivity nor relative magnetic
permeability.
Returns
-------
grad : ndarray
Adjoint-state gradient (same shape as ``simulation.model``).
"""
if self._gradient is None:
# Warn that cubic is not good for adjoint-state gradient.
if self.receiver_interpolation == 'cubic':
msg = (
"emg3d: Receiver responses were obtained with cubic "
"interpolation. This will not yield the exact gradient. "
"Change `receiver_interpolation='linear'` in the call to "
"Simulation()."
)
warnings.warn(msg, UserWarning)
# Check limitation 1: So far only isotropic models.
if self.model.case != 'isotropic':
raise NotImplementedError(
"Gradient only implemented for isotropic models."
)
# Check limitation 2: No epsilon_r, mu_r.
var = (self.model.epsilon_r, self.model.mu_r)
for v, n in zip(var, ('el. permittivity', 'magn. permeability')):
if v is not None and not np.allclose(v, 1.0):
raise NotImplementedError(
f"Gradient not implemented for {n}."
)
# Ensure misfit has been computed
# (and therefore the electric fields).
_ = self.misfit
# Compute back-propagating electric fields.
self._bcompute()
# Pre-allocate the gradient on the mesh.
gradient_model = np.zeros(self.model.grid.shape_cells, order='F')
# Loop over source-frequency pairs.
for src, freq in self._srcfreq:
# Multiply forward field with backward field; take real part.
# This is the actual Equation (10), with:
# del S / del p = iwu0 V sigma / sigma,
# where lambda and E are already volume averaged.
efield = self._dict_get('efield', src, freq)
bfield = self._dict_get('bfield', src, freq)
gfield = fields.Field(
grid=efield.grid,
data=-np.real(bfield.field * efield.smu0 * efield.field),
dtype=float,
)
# Bring the gradient back from the computation grid to the
# model grid.
gradient = gfield.interpolate_to_grid(self.model.grid)
# Pre-allocate the gradient for this src-freq.
shape = gradient.grid.shape_cells
grad_x = np.zeros(shape, order='F')
grad_y = np.zeros(shape, order='F')
grad_z = np.zeros(shape, order='F')
# Map the field to cell centers times volume.
cell_volumes = gradient.grid.cell_volumes.reshape(
shape, order='F')
maps.interp_edges_to_vol_averages(
ex=gradient.fx, ey=gradient.fy, ez=gradient.fz,
volumes=cell_volumes,
ox=grad_x, oy=grad_y, oz=grad_z)
grad = grad_x + grad_y + grad_z
# => Frequency-dependent depth-weighting should go here.
# Add this src-freq gradient to the total gradient.
gradient_model -= grad
# => Frequency-independent depth-weighting should go here.
# Apply derivative-chain of property-map
# (only relevant if `mapping` is something else than conductivity).
self.model.map.derivative_chain(
gradient_model, self.model.property_x)
self._gradient = gradient_model
return self._gradient[:, :, :self._input_sc2]
@property
def misfit(self):
r"""Misfit or cost function.
The data misfit or weighted least-squares functional using an
:math:`l_2` norm is given by
.. math::
:label: misfit
\phi = \frac{1}{2} \sum_s\sum_r\sum_f
\left\lVert
W_{s,r,f} \left(
\textbf{d}_{s,r,f}^\text{pred}
-\textbf{d}_{s,r,f}^\text{obs}
\right) \right\rVert^2 \, ,
where :math:`s, r, f` stand for source, receiver, and frequency,
respectively; :math:`\textbf{d}^\text{obs}` are the observed electric
and magnetic data, and :math:`\textbf{d}^\text{pred}` are the synthetic
electric and magnetic data. As of now the misfit does not include any
regularization term.
The data weight of observation :math:`d_i` is given by :math:`W_i =
\varsigma^{-1}_i`, where :math:`\varsigma_i` is the standard deviation
of the observation, see
:attr:`emg3d.surveys.Survey.standard_deviation`.
.. note::
You can easily implement your own misfit function (to include,
e.g., a regularization term) by monkey patching this misfit
function with your own::
@property # misfit is a property
def my_misfit_function(self):
'''Returns the misfit as a float.'''
if self._misfit is None:
self.compute() # Ensures fields are computed.
# Computing your misfit...
self._misfit = your misfit
return self._misfit
# Monkey patch simulation.misfit:
emg3d.simulation.Simulation.misfit = my_misfit_function
# And now all the regular stuff, initiate a Simulation etc
simulation = emg3d.Simulation(survey, grid, model)
simulation.misfit
# => will return your misfit
# (will also be used for the adjoint-state gradient).
Returns
-------
misfit : float
Value of the misfit function.
"""
if self._misfit is None:
# Ensure efields are computed
self.compute()
# Check if weights are stored already. (weights are currently
# simply 1/std^2; but might change in the future).
if 'weights' not in self.data.keys():
# Get standard deviation, raise warning if not set.
std = self.survey.standard_deviation
if std is None:
raise ValueError(
"Either `noise_floor` or `relative_error` or both "
"must be provided (>0) to compute the "
"`standard_deviation`. It can also be set directly "
"(same shape as data). The standard deviation is "
"required to compute the misfit."
)
# Store weights
self.data['weights'] = std**-2
# Calculate and store residual.
residual = self.data.synthetic - self.data.observed
self.data['residual'] = residual
# Get weights, calculate misfit.
weights = self.data['weights']
self._misfit = np.sum(weights*(residual.conj()*residual)).real/2
return self._misfit
def _bcompute(self):
"""Compute bfields asynchronously for all sources and frequencies."""
# Initiate back-propagated electric field and info dicts.
if not hasattr(self, '_dict_bfield'):
self._dict_bfield = self._dict_initiate
self._dict_bfield_info = self._dict_initiate
# Create iterable from src/freq-list to call the process_map.
def collect_bfield_inputs(inp):
"""Collect inputs."""
source, freq = inp
data = {
'model': self.model,
'sfield': self._get_rfield(source, freq),
# bfield is None unless it was explicitly set.
'efield': self._dict_get('bfield', source, freq),
'solver_opts': self.solver_opts
}
return self._data_or_file('bfield', source, freq, data)
# Use process_map to compute fields in parallel.
out = utils._process_map(
solver._solve,
list(map(collect_bfield_inputs, self._srcfreq)),
max_workers=self.max_workers,
**{'desc': 'Back-propagate ', **self._tqdm_opts},
)
# Loop over src-freq combinations to extract and store.
for i, (src, freq) in enumerate(self._srcfreq):
# Store bfield and solver info.
self._dict_bfield[src][freq] = out[i][0]
self._dict_bfield_info[src][freq] = out[i][1]
# Print solver info.
self.print_solver_info('bfield', verb=self.verb)
def _get_rfield(self, source, frequency):
"""Return residual source field for given source and frequency."""
freq = self.survey.frequencies[frequency]
# Get values for this source and frequency.
grid = self.get_grid(source, frequency)
residual = self.data.residual.loc[source, :, frequency].data
weight = self.data.weights.loc[source, :, frequency].data
# Initiate empty residual source field.
rfield = fields.Field(grid, frequency=freq)
# Residual source strength: Weighted residual, normalized by -smu0.
strength = np.conj(residual * weight / -rfield.smu0)
# Loop over receivers, input as source.
for i, rec in enumerate(self.survey.receivers.values()):
# Skip if no data.
if np.isnan(residual[i]):
continue
# Get absolute coordinates as fct of source.
# (Only relevant in case of "relative" receivers.)
coords = rec.coordinates_abs(self.survey.sources[source])
# Create adjoint source.
src = rec._adjoint_source(coords, strength=strength[i])
# Get and add this source field.
rfield.field += src.get_field(grid=grid, frequency=freq).field
return rfield
[docs] @utils._requires('discretize')
def jvec(self, vector):
r"""Compute the sensitivity times a vector.
.. math::
:label: jvec
J v = P A^{-1} G v \ ,
where :math:`v` has size of the model.
Parameters
----------
vector : ndarray
Shape of the model.
Returns
-------
jvec : ndarray
Shape of the data.
"""
# Ensure misfit has been computed
# (and therefore the electric fields).
_ = self.misfit
# Create iterable from src/freq-list to call the process_map.
def collect_gfield_inputs(inp, vector=vector):
"""Collect inputs."""
source, freq = inp
# Forward electric field
efield = self._dict_get('efield', source, freq)
# Compute gvec = G * vector (using discretize)
gvec = efield.grid.get_edge_inner_product_deriv(
np.ones(efield.grid.n_cells)
)(efield.field) * vector.ravel()
# Extension for tri-axial anisotropy is trivial:
# gvec = mesh.get_edge_inner_product_deriv(
# np.ones(mesh.n_cells)*3)(efield.field) * vector
gfield = fields.Field(
grid=efield.grid,
data=-efield.smu0*gvec, # -iwu: To get complete source field.
frequency=efield.frequency
)
data = {
'model': self.model,
'sfield': gfield,
'efield': None,
'solver_opts': self.solver_opts,
}
return self._data_or_file('gfield', source, freq, data)
# Compute and return A^-1 * G * vector
out = utils._process_map(
solver._solve,
list(map(collect_gfield_inputs, self._srcfreq)),
max_workers=self.max_workers,
**{'desc': 'Compute jvec ', **self._tqdm_opts},
)
# Initiate jvec data with NaN's if it doesn't exist.
if 'jvec' not in self.data.keys():
self.data['jvec'] = self.data.observed.copy(
data=np.full(self.survey.shape, np.nan+1j*np.nan))
# Loop over src-freq combinations to extract and store.
for i, (src, freq) in enumerate(self._srcfreq):
gfield = self._load(out[i][0], 'efield')
resp = self._get_responses(src, freq, gfield)
self.data['jvec'].loc[src, :, freq] = resp
return self.data['jvec'].data
[docs] def jtvec(self, vector):
r"""Compute the sensitivity transpose times a vector.
If ``vector=residual*weights``, ``jtvec`` corresponds to the
``gradient``.
.. math::
:label: jtvec
J^H v = G^H A^{-H} P^H v \ ,
where :math:`v` has size of the data.
Parameters
----------
vector : ndarray
Shape of the data.
Returns
-------
jtvec : ndarray
Adjoint-state gradient for the provided vector; shape of the model.
"""
# Note: The entire chain `gradient`->`_bcompute`->`_get_rfield` and
# also `_jtvec` should be re-factored.
# Replace residual by provided vector
# (division by weight is undone in gradient).
self.data.residual[...] = vector/self.data.weights.data
# Reset gradient, so it will be computed.
self._gradient = None
for name in ['_dict_bfield', '_dict_bfield_info']:
if hasattr(self, name):
delattr(self, name)
# Get gradient with `v` as residual.
return self.gradient
# UTILS
@property
def _dict_initiate(self):
"""Return a dict of the structure `dict[source][freq]=None`."""
return {src: {freq: None for freq in self.survey.frequencies}
for src in self.survey.sources.keys()}
@property
def _srcfreq(self):
"""Return list of all source-frequency pairs."""
if getattr(self, '__srcfreq', None) is None:
self.__srcfreq = list(
itertools.product(
self.survey.sources.keys(), self.survey.frequencies.keys()
)
)
return self.__srcfreq
def _freq_inp2key(self, frequency):
"""Return key of frequency entry given its key or its value. """
if not isinstance(frequency, str):
if not hasattr(self, '__freq_inp2key'):
self.__freq_inp2key = {
float(v): k for k, v in self.survey.frequencies.items()
}
frequency = self.__freq_inp2key[frequency]
return frequency
@property
def _info_grids(self):
"""Return a string with "min {- max}" grid size."""
# Single grid for all sources and receivers.
if self.gridding in ['same', 'single', 'input']:
grid = self.get_grid(*self._srcfreq[0])
min_nc = grid.n_cells
min_vc = grid.shape_cells
has_minmax = False
# Source- and/or frequency-dependent grids.
else:
min_nc = 1e100
max_nc = 0
# Loop over all grids and get smallest/biggest values.
for src, freq in self._srcfreq:
grid = self.get_grid(src, freq)
if grid.n_cells > max_nc:
max_nc = grid.n_cells
max_vc = grid.shape_cells
if grid.n_cells < min_nc:
min_nc = grid.n_cells
min_vc = grid.shape_cells
has_minmax = min_nc != max_nc
# Assemble info.
info = f"{min_vc[0]} x {min_vc[1]} x {min_vc[2]} ({min_nc:,})"
if has_minmax:
info += f" - {max_vc[0]} x {max_vc[1]} x {max_vc[2]} ({max_nc:,})"
return info
[docs] def print_grid_info(self, verb=1, return_info=False):
"""Print info for all generated grids."""
def get_grid_info(src, freq):
"""Return grid info for given source and frequency."""
grid = self.get_grid(src, freq)
out = ''
if verb != 0 and hasattr(grid, 'construct_mesh_info'):
out += grid.construct_mesh_info
out += grid.__repr__()
return out
# Act depending on gridding:
out = ""
# Frequency-dependent.
if self.gridding == 'frequency':
for freq in self.survey.frequencies.values():
out += f"= Source: all; Frequency: {freq} Hz =\n"
out += get_grid_info(self._srcfreq[0][0], freq)
# Source-dependent.
elif self.gridding == 'source':
for src in self.survey.sources.keys():
out += f"= Source: {src}; Frequency: all =\n"
out += get_grid_info(src, self._srcfreq[0][1])
# Source- and frequency-dependent.
elif self.gridding == 'both':
for src, freq in self._srcfreq:
out += f"= Source: {src}; Frequency: "
out += f"{self.survey.frequencies[freq]} Hz =\n"
out += get_grid_info(src, freq)
# Single grid.
else:
out += "= Source: all; Frequency: all =\n"
out += get_grid_info(self._srcfreq[0][0], self._srcfreq[0][1])
if return_info:
return out
elif out:
print(out)
[docs] def print_solver_info(self, field='efield', verb=1, return_info=False):
"""Print solver info."""
# If not verbose, return.
if verb < 0:
return
# Get info dict.
out = ""
# Loop over sources and frequencies.
for src, freq in self._srcfreq:
cinfo = self._dict_get(f"{field}_info", src, freq)
# Print if verbose or not converged.
if cinfo is not None and (verb > 0 or cinfo['exit'] != 0):
# Initial message.
if not out:
out += "\n"
if verb > 0:
out += f" - SOLVER INFO <{field}> -\n\n"
# Source and frequency info.
out += f"= Source {src}; Frequency "
out += f"{self.survey.frequencies[freq]} Hz ="
# Print log depending on solver and simulation verbosities.
if verb == 0 or self.solver_opts['verb'] != 1:
out += f" {cinfo['exit_message']}\n"
if verb == 1 and self.solver_opts['verb'] == 1:
out += f" {cinfo['log'][12:]}"
if verb == 1 and self.solver_opts['verb'] > 1:
out += f"\n{cinfo['log']}\n"
if return_info:
return out
elif out:
print(out)
def _set_model(self, model, kwargs):
"""Set self.model and self.gridding_opts."""
# Store original input_sc2. Undocumented.
# This should eventually be replaced by an `active_cells` mask.
self._input_sc2 = kwargs.pop('_input_sc2', model.grid.shape_cells[2])
# Get gridding_opts from kwargs.
gridding_opts = kwargs.pop('gridding_opts', {})
# If 'dict', entire dict should be provided.
if self.gridding == 'dict':
self._dict_grid = gridding_opts
# If 'input', TensorMesh should be provided
elif self.gridding == 'input':
self._grid_single = gridding_opts
# If 'same', there shouldn't be any options.
elif self.gridding == 'same':
if gridding_opts:
msg = "`gridding_opts` is not permitted if `gridding='same'`."
raise TypeError(msg)
# If 'source', 'frequency', 'both', 'single' => automatic gridding.
else:
g_opts = gridding_opts.copy()
# Expand model by water and air if required.
expand = g_opts.pop('expand', None)
if expand is not None:
try:
interface = g_opts['seasurface']
except KeyError as e:
msg = ("`g_opts['seasurface']` is required if "
"`g_opts['expand']` is provided.")
raise KeyError(msg) from e
model = models.expand_grid_model(model, expand, interface)
# Get automatic gridding input.
# Estimate the parameters from survey and model if not provided.
gridding_opts = meshes.estimate_gridding_opts(
g_opts, model, self.survey, self._input_sc2)
self.gridding_opts = gridding_opts
self.model = model
# DEPRECATED, will be removed from v1.4.0 onwards.
def expand_grid_model(model, expand, interface):
"""Deprecated, moved to models: `emg3d.models.expand_grid_model`."""
msg = ("emg3d: `expand_grid_model` moved from `simulations` to `models`. "
"Its availability in `simulations` will be removed in v1.4.0.")
warnings.warn(msg, FutureWarning)
return models.expand_grid_model(model, expand, interface)
def estimate_gridding_opts(gridding_opts, model, survey, input_sc2=None):
"""Deprecated, moved to meshes: `emg3d.meshes.estimate_gridding_opts`."""
msg = ("emg3d: `estimate_gridding_opts` moved from `simulations` to "
"`meshes`. Its availability in `simulations` will be removed in "
"v1.4.0.")
warnings.warn(msg, FutureWarning)
return meshes.estimate_gridding_opts(
gridding_opts, model, survey, input_sc2)