Inversion Transforming the apparent to « real » resistivity. Find a numerical model that explains the field measurment.

Direct prob. / Inverse problem Measure or data Parameters or model d m

Problem to solve Error on measurment, sub sampling Field constraints Miss choose of relevant parameters Over simplified physical model or « law » Non unicity of the solution A priori knowledge to be included Cost in time, money …..

Search for solution Minimise the error (y) between the measred data (d) and the reproduction of these data ( đ ) from a synthetic model (m): Least square (norm L 2 ) : Robust inversion (norm L 1 ) : (filtering ouliers)

The mean square approach  Linear case : solution : Non linear case : Gauss-Newton method J = jacobian matrix : Initial model :

Gauss-Newton / Quasi Gauss-Newton

Damped inversion « damped LS » (Marquardt-Levenberg or Ridge regression) : « Smoothness constraint » : C : « smoothing matrix »

Damping factors

Initial damping factor : – minimum damping factor : (valeurs par défaut) Initial damping factor : – minimum damping factor : (inversion non amortie)

Smoothness constraint NO YES

« Blocky » inversion Taking into account sharpe changes of resistivity in the model : R m and R d : matrix ginving an independant weigth of data and model in the inversion processus.

Initial model

Model discretization for forward modelling

Model discretization

1&2 1&

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