Image appraisal for 2-D and 3-D electromagnetic inversion

Linearized methods are presented for appraising resolution and parameter accuracy in images generated with 2-D and 3-D nonlinear electromagnetic (EM) inversion schemes. When direct matrix inversion is used, the model resolution and a posteriori model covariance matrices can be calculated readily. By analyzing individual columns of the model resolution matrix, the spatial variation of the resolution in the horizontal and vertical directions can be estimated empirically. Plotting the diagonal of the model covariance matrix provides an estimate of how errors in the inversion process, such as data noise and incorrect a priori assumptions, map into parameter error and thus provides valuable information about the uniqueness of the resulting image. Methods are also derived for image appraisal when the iterative conjugate gradient technique is applied to solve the inverse. An iterative statistical method yields accurate estimates of the model covariance matrix as long as enough iterations are used. Although determining the entire model resolution matrix in a similar manner is computationally prohibitive, individual columns of this matrix can be determined. Thus, the spatial variation in image resolution can be determined by calculating the columns of this matrix for key points in the image domain and then interpolating between. Examples of the image analysis techniques are provided on 2-D and 3-D synthetic cross-well EM data sets as well as a field data set collected at Lost Hills oil field in central California.