(mesh-design)=

# Mesh design

An appropriately designed computational mesh is an important component
to the geophysical inversion. There are both general and survey-specific
strategies that may be used to design 'good' meshes for inversion. This
section provides complementary details to the core [Octree
creation](https://mirageoscience-octree-creation-app.readthedocs-hosted.com/en/latest)
documentation, so readers may want to reference both.

## Base parameters

An Octree mesh can be thought of as the combination of a base mesh and
refinements. The base mesh is defined by the core extent, padding, and
the base cell resolution. These concepts are explored in further detail
in the following subsections.

- [A region of interest (extent)](mesh-extent)
- [Base cell resolution](base-cell-size)
- [Padding distance](padding-distance)

![base_mesh](./images/mesh_core.png)

(mesh-extent)=

### Mesh extent

The mesh extent is provided as a `geoh5py.object` from which the
bounding box is computed. For geophysical inversion this will in most
cases be the geophysical survey object so that core region of the mesh
will be centered below the data. For all `geoh5py.BaseSurvey` objects
that contain a `complement` object (transmitters, base stations, current
electrodes), the extent will be computed from the superposition of the
object and its complement.

(base-cell-size)=

### Base cell size

Next, we need to define a base cell size for the core region. A few
criteria need to be taken into consideration:

- Sources/receivers separation
- Terrain clearance (height above ground)
- Modeling resolution (size of the target)

Numerical artefacts due to the discretization of topography decay
sufficiently when receivers are roughly two cell dimensions away from
the nearest edge or corner of a mesh cell in the air/earth boundary. For
example, if the lowest drape height in given airborne survey is 60 m, a
prudent cell size would be in the range of \~30 m. Another rule of thumb
is that the cell size should be smaller than half the smallest data
spacing. This is especially important for electrical methods where the
electric field may be discontinuous and numerical artefacts may arise if
the cells are not small enough to capture rapid changes in the field.
For example, a 40m dipole direct-current survey would suggest a cell
size around \~20m.

(padding-distance)=

### Padding cells

As a general rule of thumb, the padding region should be at least as
wide as the data extent in order to easily model features with
wavelengths that may extend beyond the surveyed area.

In the case of EM modeling, we also need to consider the diffusion
distance of the EM fields. The [skin
depth](http://em.geosci.xyz/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/skindepth.html?highlight=skin%20depth#approximations)
represents the distance over which the fields decay by a factor of
$1/e$. The skin depth can thus be used to add a padding distance that
incorporates the minimum area within which the fields have any influence
on the solution. The skin depth can be calculated for the
frequency-domain system by:

$$\delta = 503 \sqrt{\frac{\rho}{f}}$$

where $\rho$ is the expected resistivity of the background and $f$ is
the base frequency of the system.

Equivalently, for time-domain systems

$$\delta = \sqrt{\frac{2t \; \rho}{\mu_0}} \approx 1261 \sqrt{t \; \rho}$$

where $t\;(sec)$ is the largest time measured by the system,
$\rho \; (\Omega.m)$ is the expected resistivity of the background and
$\mu_0 \; (4 \pi * 1e-7)$ is the permeability of free space.

## Refinements

An octree mesh can be refined in specific regions - warranted for either
numerical accuracy or modeling purposes. Fine cells can be "added" to
the octree grid using various insertion methods that depend on the type
of object used:

- Radial: Add concentric spherical shells of cells around points. It is
  the default behaviour for `Points`-like objects.
- Line: Add concentric tubes around line segments. It is the default
  discretization for line sources (EM loops) and `Curve` objects.
- Surface: Add layers of cells along triangulated surfaces in 3D. It is
  the default behaviour for `Surface` objects and mostly used to refine
  known boundaries between geological domain and topography.
- Horizon \[Optional\]: Add horizontal layers of cells below a
  triangulated surface computed from the input object's vertices. This
  is a useful option to create a core region below the survey area.

See the [Octree-Creation-App:
Refinements](https://mirageoscience-octree-creation-app.readthedocs-hosted.com/en/latest/methodology.html#refinements)
section for more details.

### For direct-current survey

For EM methods, such as electrical direct-current surveys, the numerical
accuracy of the forward simulation is the primary factor to decide on a
mesh refinement strategy. It is important to have several small cells
around each source and receivers for solving partial differential
equations accurately. In this case, dipole source and receivers must be
converted to a `Points` object in order to trigger a `radial`
refinement. Concentric shells of cells are added around each
source/receiver poles.

![dc_mesh](./images/dc_mesh_diagram.png)

A second level of `Surface` refinement was added along the triangulated
topography. Only coarse cells (3th octree level) were needed to preserve
good continuity of the fields near the boundary of the domain.

### For gravity and magnetic surveys

For potential field methods, the forward simulation is achieved via
integration over prisms (linear operator) which is much less sensitive
to the choice of discretization. Moreover, only cells below topography
(active cells) are considered. For a more optimal mesh design, only fine
cells are required at the air-ground interface. A `horizon` style
refinement was used in order to add layers of fine cells below the
footprint of the gravity and magnetic sensors. The maximum `distance`
parameter was set to 100 m to reduce the number of small cells away from
the receiver locations.

![mag_mesh](./images/mag_mesh_diagram.png)

## Auto-mesh

The mesh can be generated in an automatic fashion where the meshing
parameters are estimated from the geometry of survey and topography. The
mesh parameter in the SimPEG ui.json files is optional. If the user
leaves the mesh empty, a mesh will be created during setup for the
subsequent inversion. This section describes how the meshing parameters
are estimated for the auto-mesh routine.

### Core parameters

The auto-mesh routine refines radially about the survey locations and
below the topography surface. The refinement for the survey creates
three concentric shells of four cells at each level. The topography
refinement places a single layer of two cells in the third octree level
below the earth-air interface. The minimum level is set to six adding a
base refinement to the whole mesh so that the cells in the padding
region do not coarsen excessively.

### Cell size

The cell size will be chosen as half the median station spacing of the
survey.

![cell_size](./images/cell_size_estimate.png)

### Padding

The padding size can be estimated from the extent of the survey object.
The auto-mesh will first compute the horizontal extent and take half of
the largest extent (in x/y) for the horizontal padding parameters.

![extent_to_pad](./images/extent_to_pad.png)