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