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. The 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 boreholes. For computational efficiency a localized nonlinear approximation is employed to compute the sensitivity matrix. Parameter variance estimates are determined using an iterative Monte Carlo technique that assumes the data contain measurement noise, and that constraint assumptions imposed on the model are in error. The a posteriori model covariance matrix is determined statistically for the linearized technique by rerunning the last iteration of the nonlinear inversion N times, each time adding random errors to the data and constraints. The nonlinear approach involves rerunning the full inversion N times. Two oil field examples from California indicate that the linearized approach produces the same general pattern in the uncertainty estimates as the nonlinear estimation process. However, the linearized estimates are smaller in magnitude and show less spatial variation compared to the full nonlinear estimates, and the deviation between the two techniques increases as the contrast between the maximum and minimum conductivities within the inversion domain becomes greater. (Some figures in this article are in colour only in the electronic version; see www.iop.org)