Methods are presented for appraising resolution and uncertainty in images generated with large-scale nonlinear EM inversion schemes where singular value decomposition or direct matrix inversion is not possible. The methods explore the computation of the model resolution matrix (MRM) and model covariance matrix (MCM) using a conjugate gradient (CG) method. The proposed methods enable the computation of the MRM and MCM for all the inversion iterations without considerable sacrifices in the total computation time. Examples of the resolution and uncertainty analysis are provided for the inversion of Marine CSEM and cross-well EM synthetic and field data.
Electromagnetic (EM) technologies, such as marine controlled source EM (CSEM) and cross-well EM, are becoming commonly used geophysical tools in petroleum exploration and reservoir fluid monitoring. It is important to appraise the images that arise in the inversion of EM data for quality control. Existing image appraisal techniques for 2-D and 3-D EM inversion are based on the calculation of Model Resolution Matrix and Model Covariance Matrix (Menke, 1984; Alumbaugh and Newman, 2000). Although these quantities are linearized at any iteration of a non-linear inversion, they provide important information about how good and reliable the images are. The model resolution matrix,
defines the connection between the true model,
and the model estimated via non-linear least-squares inversion (
ˆ ) as follows
ˆ =
×
(1) Ideally, when all model components are perfectly resolved, the
is equal to the identity matrix, which implies perfect resolution. If the model is not perfectly constrained,
will depart from being the identity matrix, which means that the model contains some null-space components that are not constrained by the data. The Model Resolution Matrix can be interpreted differently depending which part of the
is used. The rows of
can be treated as averaging functions, which means that the estimates of the model parameters are really weighted averages of the true model parameters if
is not an identity matrix. In other words, the rows of the Model Resolution Matrix describe what scale of features in the model can be actually resolved. The columns of
can be treated as Backus-Gilbert point-spread functions (PSF) (Alumbaugh and Newman, 2000) which describe the resolution at the point of interest in the image. The distribution of the main diagonal elements of the Model Resolution Matrix can be a good indicator of the image resolution. The more that
deviates from the identity matrix, the less resolution the image will have. The Model Covariance Matrix characterizes the degree of error amplification that occurs in the mapping from data to model parameters. The error can be data noise, bias, and inappropriate a priori assumptions about the model. In this sense, it serves an important role for uncertainty analysis for the model parameters.
Nonlinear least-squares inversion usually ends up with solving a linearized equation for each iteration (e.g. see Abubakar et al, 2005). For large-scale problems, direct matrix inversion becomes very expensive and impractical.
