(data-misfit)=

# Data Fitting

The data misfit function ($\phi_d$) measures the quality of a model
$\mathbf{m}$ at **fitting the observed data**. It generally takes the
form

$$\phi_d =\sum_{i=1}^{N}\left(\frac{d_i^{pred} - {d}_i^{obs}}{w_i}\right)^2 \;.$$

It measures the sum of **weighted residual** between observed
($\mathbf{d}^{obs}$) and predicted ($\mathbf{d}^{pred}$) data. In matrix
form, this least-squares measure can be written as

$$\phi_d =(F(\mathbf{m}) - \mathbf{d}^{obs})^T \mathbf{W}_d^T \mathbf{W}_d (F(\mathbf{m}) - \mathbf{d}^{obs})\;,$$

where $F(\mathbf{m})$ is a forward simulation of data (gravity,
magnetic, EM, etc.) due to a model $m$. The $\mathbf{W}_d$ is a diagonal
matrix holding the data weights ($\frac{1}{\mathbf{w}}$), also referred
to as **data uncertainties**. The larger the weights, the less important
a datum is to drive the solution. There are many sources of **noise**
that can yield large data residuals:

- Experimental noise from $\mathbf{d}^{obs}$, generally introduced
  during acquisition (positioning, instrumental noise)
- Numerical noise introduced by $F(\mathbf{m})$ from our inability to
  perfectly simulate the physics (discretization, interpolation, etc.)

In all cases, we may never expect the data residual to vanish
completely, but we may allow it to be within an *acceptable* level of
fitness for modelling purposes.

## Uncertainties

As mentioned in the `` `Data Misfit ``{.interpreted-text
role="std:ref"}\` section, we want to assign **data uncertainties** that
reflect our inability to perfectly replicate the observed data, within
an acceptable level. While many strategies can be used, we are going to
demonstrate basic principles to assess the effectiveness of a chosen
method. We will employ two basic strategies:

- [Floor](constant-floor) value set as a constant,
- [Amplitude-based (%)](amplitude-perct) values with variable weights.

We are going to use the synthetic magnetic example shown below. We added
20 nT of random Gaussian noise to the simulated data to emulate field
conditions.

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

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

(constant-floor)=

### Constant floor

The simplest choice is to assign a constant value across the entire
dataset - also referred to as **"floor"** uncertainties. It is related
to the expected standard deviation of a Gaussian distribution if the
noise is random.

In a *real world* context, the true noise level is unknown. It is common
practice to estimate an appropriate floor by trial and error. The
following three experiments demonstrate typical characteristics to be
expected when attempting to determine an optimal noise floor.

#### Noise floor too high

With a high floor, the inversion is dominated by the
[regularization](regularization) function. As the data remains
underfitted, a coherent signal remains unaccounted for. The model is
very smooth, and we have a poor recovery of the shape of the susceptible
body.

![large_floor_plot](./images/large_floor_plot.png)
![large_floor](./images/large_floor.png)

#### Noise floor too low

With this inversion, the uncertainties are intentionally set lower than
the known random noise level. The inversion attempts to fit both the
signal as well as the random noise. This gives rise to a spiky model
with a lot of unwanted structure in the surface. The inversion also ran
many more iterations in order to suppress the regularization function.

![small_floor_plot](./images/small_floor_plot.png)
![small_floor](./images/small_floor.png)

#### Optimal

The following inversion is with a well-chosen floor uncertainty. The
model is fitting the data well without fitting the background noise. The
model is smooth but retains the shape of the source.

![correct_floor_plot](./images/correct_floor_plot.png)
![correct_floor](./images/correct_floor.png)

(amplitude-perct)=

### Amplitude (%)

The second common strategy is to assign uncertainties based on the
strength of the signal. In this scenario, we have assigned 10% of the
amplitude of data plus a small floor (2 nT) to avoid singularity near
zero:

$$\mathbf{w} = \frac{1}{|\mathbf{d}^{obs}| + \epsilon}$$

The figures below show the predicted data and recovered model after
convergence of the inversion. We note that the predicted data fit well
in the background, but underfit directly over the main anomaly. This is
generally not desirable as the important portion of the geophysical
signal is in the large amplitudes. The poor fit is reflected in the
recovered low susceptibility values and smooth shape of the magnetic
body.

![percent_floor_plot](./images/percent_floor_plot.png)
![percent_floor](./images/percent_floor.png)

## Summary

In this section we have demonstrated the effect uncertainties have on
the outcome of an inversion, both on the data residuals and on the
recovered model. It generally takes several inversions to determine
appropriate uncertainties and involves some level of subjectivity from
the user.