# 6.9. Data Weighting and Joint Inversion¶

Joint inversion inverts two or more datasets simultaneously to recover a unifying physical property model. For example, any combination of MT, ZTEM, FDEM or TDEM data may be inverted jointly to recover a conductivity model. When performing joint inversion, it is important that the data misfit for each individual dataset is appropriately weighted. If this is not done correctly, the inversion will overfit a particular dataset at the expense of others; typically the dataset with the largest number of data observations.

To weight the data appropriately, we let the total data misfit for the inversion ($$\phi_d$$) be the weighted sum of the data misfits for each dataset, i.e.:

(1)$\phi_d = c_1 \phi_d^{(1)} + c_2 \phi_d^{(2)} + \, ... \; = \sum_{k=1}^K c_k \phi_d^{(k)}$

where $$K$$ is the total number of datasets, $$c_k$$ are the weighting constants and:

(2)$\phi_d^{(k)} = \big \| \mathbf{W_d^{(k)}} \big ( \mathbf{d_{obs}^{(k)}} - \mathbb{F^{(k)}}[\boldsymbol{\sigma}] \big ) \big \|^2$

Thus for dataset $$k$$:

• $$\mathbf{W_d^{(k)}}$$ is a diagonal matrix containing the reciprocal of the data uncertainties

• $$\mathbf{d_{obs}^{(k)}}$$ is the set of field observations

• $$\mathbb{F^{(k)}}$$ denotes the forward modeling operator.

## 6.9.1. Weight by number of data¶

Using this approach, we weight each dataset based on the number of field observations. Datasets with more field observations given smaller weighting constants $$c_k$$ and visa versa. Let $$N_1, \, N_2 , \, N_3, \, ... \, , N_K$$ be the total number of data observations for datasets 1, 2, 3, … , $$K$$, respectively. Where $$\widetilde{N}$$ is the average number of data observations for all datasets, we let:

$c_k = \frac{\widetilde{N}}{N_k}$

The benefits of this approach are as follows:

1. the target data misfit for the inversion remains the same; i.e. it is the total number of data observations for all datasets

2. the inversion no longer fits datasets based on the number of data observations