Data Misfit

The data misfit using an l_2 norm is:

\[\mu_\text{data} = {1\over 2}\left| \mathbf{W}_d (\mathbf{d}_\text{pred} - \mathbf{d}_\text{obs}) \right|_2^2\]

If the field, u, is provided, the calculation of the data is fast:

\[ \begin{align}\begin{aligned}\mathbf{d}_\text{pred} = \mathbf{Pu(m)}\\\mathbf{R} = \mathbf{W}_d (\mathbf{d}_\text{pred} - \mathbf{d}_\text{obs})\end{aligned}\end{align} \]

Where P is a projection matrix that brings the field on the full domain to the data measurement locations; u is the field of interest; d_obs is the observed data; and \(\mathbf{W}_d\) is the weighting matrix.

The derivative of this, with respect to the model, is:

\[\frac{\partial \mu_\text{data}}{\partial \mathbf{m}} = \mathbf{J}^\top \mathbf{W}_d \mathbf{R}\]

The second derivative is:

\[\frac{\partial^2 \mu_\text{data}}{\partial^2 \mathbf{m}} = \mathbf{J}^\top \mathbf{W}_d \mathbf{W}_d \mathbf{J}\]

The API

class SimPEG.DataMisfit.BaseDataMisfit(survey, **kwargs)

Note

You should inherit from this class to create your own data misfit term.

Wd
counter = None
debug = False
nP

Common Data Misfits

l2 norm

class SimPEG.DataMisfit.l2_DataMisfit(survey, **kwargs)

The data misfit with an l_2 norm:

\[\mu_\text{data} = {1\over 2}\left| \mathbf{W}_d (\mathbf{d}_\text{pred} - \mathbf{d}_\text{obs}) \right|_2^2\]
W

The data weighting matrix.

The default is based on the norm of the data plus a noise floor.

Return type:scipy.sparse.csr_matrix
Returns:W
deriv(m, f=None)

Derivative of the data misfit

\[\mathbf{J}^{ op} \mathbf{W}^{ op} \mathbf{W} (\mathbf{d} - \mathbf{d}^{obs})\]
Parameters:
deriv2(m, v, f=None)
\[\mathbf{J}^{ op} \mathbf{W}^{ op} \mathbf{W} \mathbf{J}\]
Parameters:
eps = None
eps_factor = 1e-05
std = 0.05