Linearized and nonlinear parameter variance estimation for two-dimensional. electromagnetic induction inversion

Linearized and nonlinear techniques are presented for determining estimates of parameter uncertainty within a two-dimensional iterative Born scheme. T he scheme employs low frequency (< 100 kHz) magnetic dipole sources in one well, and uses measurements of the vertical magnetic field in a second well to invert for the electrical conductivity distribution between the two bor eholes. For computational efficiency a localized nonlinear approximation is employed to compute the sensitivity matrix. Parameter variance estimates a re determined using an iterative Monte Carlo technique that assumes the dat a contain measurement noise, and that constraint assumptions imposed on the model are in error. The ri posteriori model covariance matrix is determine d statistically fur the linearized technique by rerunning the last iteratio n of the nonlinear inversion N times, each time adding random errors to the data and constraints. The nonlinear approach involves rerunning the full i nversion N times, Two oil field examples from California indicate that the linearized approach produces the same general pattern in the uncertainty es timates as the nonlinear estimation process.. However, the linearized estim ates are smaller in magnitude and show less spatial variation compared to t he full nonlinear estimates, and the deviation between the two techniques i ncreases as the contrast between the maximum and minimum conductivities wit hin the inversion domain becomes greater.