# Depth of Investigation

This application allows users to compute a depth of investigation from
model values. In geophysics, "depth of investigation" refers to the
maximum depth below the surface from which a geophysical survey can
reliably measure. It depends on factors like the survey design and
physical properties of the subsurface material.

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

Several strategies have been proposed to assess uncertainties in models
recovered from inversion. {cite:t}`nabighian_1989` used a skin depth
approach for electromagnetic surveys, assuming a background halfspace
resistivity. {cite:t}`li_1999` implemented a cut-off value based on two
inverted models obtained with slightly different assumptions.

This application uses the method proposed by
{cite:t}`christiansen_2012`, based on the sum-square of sensitivities to
estimate a volume of low confidence.

## Interface

The depth of investigation methods based on sensitivity cutoffs relies
on the ui.json interface.

![interface](./images/uijson.png)

### Inputs

- **Mesh**: The mesh used for inversion
- **Sensitivity**: Model selector for the sensitivities at a given
  iteration (see [Pre-requisite](pre-requisite))
- **Cut-off**: Percentage value to threshold the sensitivities, relative
  to the **Method** used.
- **Method**: One of *percentile*, *percent* or *log percent*

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

### Output

After running, the application will create a masking attribute saved on
the mesh.

![mask](./images/sensitivity_mask.png)

The mask can then be applied to any of the iterations to show only the
cells that exceeded the sensitivity threshold.

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

(pre-requisite)=

### Pre-requisite

In order to save the sensitivities during a SimPEG inversion, the 'Save
sensitivities' option must be selected from the 'Optional parameters'
tab of an inversion.

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

This will result in a new model generated and saved into the
computational mesh at each iteration.

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

## Methodology

The sensitivity matrix is calculated as part of the optimization problem
solved by SimPEG while inverting geophysical data. The sensitivity
matrix represents the degree to which each predicted datum changes with
respect to a perturbation in each model cell. It is given in matrix form
by

$$\mathbf{J} = \frac{\mathbf\partial{F}(\mathbf{m})}{\partial{\mathbf{m}}}$$

where $\mathbf{m}$ is the model vector, and $F(\mathbf{m})$ represents
the forward modelling operation as a function of the model. The
sensitivity matrix $\mathbf{J}$ is a dense array with dimensions
$N\times M$, where $N$ is the number of data and $M$ are the number of
mesh cells.

The depth of investigation mask is a property of the cells of the mesh
only so the rows of the sensitivity matrix (data) are sum-square
normalized as follows.

$$\|\mathbf{j}\| = \left[\sum_{n=1}^{N}\left(\frac{\mathbf{J}_{n:}}{w_n}\right)^{2}\right]^{(1/2)}$$

where $w_n$ are the data uncertainties associated with the $n^{th}$
datum.

The resulting vector $\|\mathbf{j}\|$ can then be thought of as the
degree to which the aggregate data changes due to a small perturbation
in each model cell. The depth of investigation mask is then computed by
thresholding those sensitivities

### Thresholding

The depth of investigation can be estimated by assigning a threshold on
the sum-squared sensitivity vector. This can be done as a percentile,
percentage, or log-percentage. In the percentile method, the mask is
formed by eliminating all cells in which the sensitivity falls below the
lowest $n$% of the number of data where $n$ is the chosen cutoff. In the
percent method the data are transformed into a percentage of the largest
value

$$d_{scaled} = \frac{100 \cdot d}{max(d)}$$

and the mask is formed by eliminating all cells in which the sensitivity
falls below the lowest $n$% of the data values where $n$ is the chosen
cutoff. Finally, the log-percent mask transforms the data into log-space
before carrying out the percentage thresholding described above.