# Joint Surveys Inversion

The `joint surveys` inversion framework allows to invert multiple
datasets at once for a single physical property model. The goal is to
combine complementary information from various geophysical surveys that
sense the Earth differently.

```{image} images/joint_survey.svg
:scale: 200%
:align: center
```

For example, a magnetotelluric survey is mostly sensitive to deep
structures but can still be affected by local changes in resistivity
near the sensor locations. A ground direct-current survey on the other
hand is highly sensitive to near surface changes in resistivity.
Inverting both surveys together would provide complementary information
that improves our modeling capabilities overall.

## Background

The joint survey inversion does not require a coupling term - simply the
summation of multiple misfit functions influencing the same model. The
global misfit function becomes

$$\phi_d = \phi_d^A + \phi_d^B + ...$$

or

$$\phi_d = \sum_{j=A, B, C} \phi_d^j$$

Since each misfit tries to update the same model values, the partial
derivatives of each function is also a summation, such that

$$\frac{\delta \phi_d}{\delta m} = \sum_{j=A, B, C} \frac{\delta \phi_d^j}{\delta m}$$

## Interface

The joint survey inversion user requires a list of standalone inversion
groups as input.

```{image} images/joint_surveys_ui.png
:scale: 50%
:align: center
```

### Input parameters

- `Group A, B and C`

  :   Standalone inversion groups to be included in the joint inversion.
      Up to three groups can be included in the joint inversion, but
      only two are required. Each group should be fully defined and runnable as a
      standalone inversion problem, with its own survey and mesh. Any
      other regularization or optimization parameters will be ignored
      and overridden by the joint inversion framework.

- `Misfit Scales A, B and C`

  :   For each standalone inversion group, a scaling factor to be
      applied to the misfit function. This allows to scale the
      uncertainties of individual surveys.

### Inversion mesh

Optional octree mesh object used for by regularization and storing the models. If not supplied, a new mesh is generated by combining the finest cell size encountered across all standalone meshes. See details in the [inversion mesh](inversion_mesh) section.

### Advanced parameters

All advanced parameters related to the regularization and optimization of
the standalone inversions are ignored and overridden by the joint
inversion framework.

(misfit_scaling)=

#### Auto-scaling of misfit functions

By default, an auto-scaling of the misfit functions is applied at each
iteration, such that the contribution of each survey to the model update
is balanced.

```{image} images/joint_surveys_ui_advanced.png
:scale: 50%
:align: center
```

This is particularly important when the surveys have
different units or sensitivities. This auto-scaling strategy prevents
the inversion from being dominated by a single survey, or overfitting a
survey while the others are still far from their target.

The scaling factor for each survey is computed as the ratio between the
achieved chi factor to the maximum chi factor of all surveys, scaled by
the cooling factor of the trade-off parameter, such that

$$\text{scale}_j = 1 - (1 - \Delta\beta) \frac{\chi_{max} - \chi_j}{\chi_{max}}$$

where $\Delta \beta$ is the chosen cooling factor of the trade-off
parameter, and $\chi_j$ are the chi factors for each survey. Chi factors are computed as the ratio
between the data misfit and the number of data points:

$$\chi_j = \frac{\phi_d^j}{N_j} \,.$$

The misfit with the highest chi factor will have a scaling
factor of 1, while the other misfits will have a scaling factor between
$\delta \beta$ and 1, depending on how far their chi factor is from the
maximum. In other words, misfits with the lowest chi factors are scaled
down by the same amount as the trade-off parameter, effectively
cancelling the cooling step as long as their chi factor remains small. As
the inversion progresses and the chi factors of all surveys reach a
similar value, the scaling factors will increase towards 1, allowing all
misfits to progress towards their target.