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

Abstract

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.

A salt tracer test monitored with surface ERT to detect preferential flow and transport paths in fractured/karstified limestones

“…Among them, the resolution (e.g., Alumbaugh and Newman, 2000;Friedel, 2003;Oldenborger and Routh, 2009) or sensitivity (e.g., Nguyen et al, 2009) matrix analysis and/or the depth of investigation (DOI) index (e.g., Oldenburg and Li, 1999;Marescot et al, 2003) analysis are often used. We used the relative sensitivity matrix computed by the Res2Dinv software (Loke and Barker, 1996) because this parameter gives a direct indication of the sensitivity of measurements subject to changes in the electrical structures.…”

VSP traveltime inversion: Near‐surface issues

“…The off-diagonal terms are the covariances, which contain information on the relationships between parameters. As shown in Alumbaugh and Newman (2000), the model covariance matrix for a linearized inversion can be calculated using…”

Applicability of electrical resistivity tomography monitoring to coarse blocky and ice‐rich permafrost landforms

“…Appraisal methods aim to evaluate: (a) the confidence we can have in the existence of features observed in the inverted image, (b) the level of detail that can be obtained and (c) the resolution capability at depth (Oldenburg and Li, 1999). Different appraisal methods were presented and discussed by Oldenburg and Li (1999), Alumbaugh and Newman (2000), Friedel (2003), Stummer et al (2004), and Routh and Miller (2006).…”