misfit

emg3d.optimize.misfit(simulation)

Return the misfit function.

The data misfit or weighted least-squares functional using an \(l_2\) norm is given by

(36)\[ \phi = \frac{1}{2} \sum_f\sum_s\sum_r \left\{ \left\lVert W_{s,r,f} \left( \textbf{d}_{s,r,f}^\text{pred} -\textbf{d}_{s,r,f}^\text{obs} \right) \right\rVert^2 \right\} + R \ .\]

Here, \(f, s, r\) stand for frequency, source, and receiver, respectively; \(\textbf{d}^\text{obs}\) are the observed electric and magnetic data, and \(\textbf{d}^\text{pred}\) are the synthetic electric and magnetic data. Finally, \(R\) is a regularization term.

The data weight of observation \(d_i\) is given by \(W_i = \varsigma^{-1}_i\), where \(\varsigma_i\) is the standard deviation of the observation (see emg3d.surveys.Survey.standard_deviation).

Note

This is an early implementation of the misfit function. Currently not yet implemented are:

• Magnetic data;
• Regularization term.

Parameters:

simulation : emg3d.simulations.Simulation

The simulation.

Returns:

misfit : float

Value of the misfit function.