Constructing piecewise-constant models in multidimensional minimum-structure inversions

Abstract

A modification of the typical minimum-structure inver-sion algorithm is presented that generates blocky, piecewise-constant earth models. Such models are often more consistent with our real or perceived knowledge of the subsurface than the fuzzy, smeared-out models produced by current minimum-structure inversions. The modified algorithm uses [Formula: see text]-type measures in the measure of model structure instead of the traditional sum-of-squares, or [Formula: see text], measure. An iteratively reweighted least-squares procedure is used to deal with the nonlinearity introduced by the non-[Formula: see text] measure. Also, and of note here, diagonal finite differences are included in the measure of model structure. This enables dipping interfaces to be formed. The modified algorithm retains the benefits of the minimum-structure style of inversion — namely, reliability, robustness, and minimal artifacts in the constructed model. Two examples are given: the 2D inversion of synthetic magnetotelluric data and the 3D inversion of gravity data from the Ovoid deposit, Voisey’s Bay, Labrador.

Focused time-lapse inversion of radio and audio magnetotelluric data

“…Using the l 2 -norm to quantify model structure, as in Occam inversion, favors smooth transitions of model properties over a number of model cells (e.g., Farquharson, 2008). If sharp transitions between geological units or anomalies with small spatial supports are expected, it is necessary to work with other model norms to obtain models in agreement with such pre-supposed properties.…”

“…Farquharson and Oldenburg (1998) minimized an l 1 -type measure of the horizontal and vertical derivatives in the 2D inversion of electrical resistance data. Farquharson (2008) minimized an approximate l 1 -norm of a combination of horizontal and vertical model differences together with differences between diagonal cells to better image dipping structures when inverting gravity and magnetotelluric (MT) data. Pilkington (2009) used the Cauchy norm to obtain sparse 3D magnetic models.…”

Optimized arrays for 2D cross‐borehole electrical tomography surveys

“…The L1-norm method was used for both the data misfit and model roughness filters (Loke et al 2003). The roughness filter with diagonal components was used as the high resistivity blocks also have sloping sides (Farquharson 2008). A cooling sequence method was used to set the damping factor in equation (1).…”

Minimum‐structure borehole gravity inversion

“…However, the models obtained are typically of smeared shapes and do not exhibit the sharp interfaces that are usually assumed to separate subsurface geologic structures. Techniques for developing sharper interfaces in models have been described by Last and Kubik (1983) , Portniguine andZhadov (1999) , andFarquharson (2008) .…”

“…If a Gauss-Newton minimization process is used to minimize t he objective function, equations ( 45) and ( 46) are rearranged to produce t he following (Farquharson, 2008),…”