misfit

emg3d.optimize.misfit(simulation)

Return the misfit function.

The weighted least-squares functional, often called objective function or misfit function, as implemented in emg3d, is given by Equation 1 in [PlMu08],

(32)\[ J(\textbf{p}) = \frac{1}{2} \sum_f\sum_s\sum_r \left\{ \left\lVert W_{s,r,f}^e \Delta^e \right\rVert^2 + \left\lVert W_{s,r,f}^h \Delta^h \right\rVert^2 \right\} + R(\textbf{p}) \ ,\]

where \(\Delta^{\{e;h\}}\) are the residuals between the observed and synthetic data,

(33)\[ \Delta^e = \textbf{e}_{s,r,f}[\sigma(\textbf{p})] -\textbf{e}_{s,r,f}^\text{obs} \ ,\]

and

(34)\[ \Delta^h = \textbf{h}_{s,r,f}[\sigma(\textbf{p})] -\textbf{h}_{s,r,f}^\text{obs} \ .\]

Here, \(f, s, r\) stand for frequency, source, and receiver, respectively; \(W^{\{e;h\}}\) are the weighting functions for the electric and magnetic data residual, \(\{\textbf{e};\textbf{h}\}^\text{obs}\) are the observed electric and magnetic data, and \(\{\textbf{e};\textbf{h}\}\) are the synthetic electric and magnetic data, computed for a given conductivity \(\sigma\), which depends on the model parameters \(\textbf{p}\). Finally, \(R(\textbf{p})\) is a regularization term.

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.