misfit

emg3d.optimize.misfit(simulation)

Misfit or cost function.

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

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

where \(s, r, f\) stand for source, receiver, and frequency, respectively; \(\textbf{d}^\text{obs}\) are the observed electric and magnetic data, and \(\textbf{d}^\text{pred}\) are the synthetic electric and magnetic data. As of now the misfit does not include any 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

You can easily implement your own misfit function (to include, e.g., a regularization term) by monkey patching this misfit function with your own:

def my_misfit_function(simulation):
    '''Returns the misfit as a float.'''

    # Computing the misfit...

    return misfit

# Monkey patch optimize.misfit:
emg3d.optimize.misfit = my_misfit_function

# And now all the regular stuff, initiate a Simulation etc
simulation = emg3d.Simulation(survey, grid, model)
simulation.misfit
# => will return your misfit
#   (will also be used for the adjoint-state gradient).

Parameters

simulationSimulation

The simulation; a emg3d.simulations.Simulation instance.

Returns

misfitfloat

Value of the misfit function.

Optimize gradient