Invariance, groups, and non-uniqueness: The discrete case

Abstract

S U M M A R YLie group methods provide a valuable tool for examining invariance and non-uniqueness associated with geophysical inverse problems. The techniques are particularly well suited for the study of non-linear inverse problems. Using the infinitesimal generators of the group it is possible to move within the null space in an iterative fashion. The key computational step in determining the symmetry groups associated with an inverse problem is the singular value decomposition of a sparse matrix. I apply the methodology to the eikonal equation and examine the possible solutions associated with a crosswell tomographic experiment. Results from a synthetic test indicate that it is possible to vary the velocity model significantly and still fit the reference arrival times. The approach is also applied to data from crosswell surveys conducted before and after a CO 2 injection at the Lost Hills field in California. The results highlight the fact that a fault cross-cutting the region between the wells may act as a conduit for the flow of water and CO 2 .

Structure‐coupled multiphysics imaging in geophysical sciences

“…From an imaging point of view, the key sources of uncertainty are data inaccuracy and limited bandwidth, incomplete knowledge and information about the typically complex physical system under investigation, and predictive modeling (theoretical and discretization) errors. These sources of uncertainty have been treated using either deterministic or probabilistic approaches [see, e.g., DeVolder et al , 2002; Reagan et al , 2003; Caers et al , 2006; Chappell and Lancaster , 2007; Vasco , 2007; Cardiff and Kitanidis , 2009; Meju , 2009 and references therein]. A particularly difficult challenge is error estimation in the solution of partial differential equations that constitute our numerical representation of the physical system being investigated [e.g., DeVolder et al , 2002; Reagan et al , 2003].…”

“…Many approaches have been proposed to quantify this uncertainty in multidimensional inverse modeling. The deterministic approaches include linear sensitivity analysis [e.g., Alumbaugh and Newman , 2000; Kalscheuer and Pedersen , 2007; Kalscheuer et al , 2010] and construction of extremal solutions [e.g., Vasco , 2007; Meju , 2009]. The probabilistic approaches include density function estimation [ Snieder , 2004; Caers et al , 2006; Cardiff and Kitanidis , 2009] and covariance estimation by Monte Carlo integration [e.g., Alkhatib and Schuh , 2007].…”

On the use of sensitivity tests in seismic tomography

“…Within a linear or weakly nonlinear framework, a number of different techniques have been tried, including multiple starting models (Vasco et al 1996), the so-called 'null space shuttle' (Deal & Nolet 1996;de Wit et al 2012), regularized extremal bounds analysis (Meju 2009), Lie group methods (Vasco 2007) and the dynamic objective function scheme (Rawlinson & Kennett 2008). However, it is in the realm of fully nonlinear sampling where the greatest strides are currently being made.…”

Location of seismic events using inaccurate data from very sparse networks

“…I will therefore avoid a wording that implies vectors and vector spaces (like 'null space' to denote non-uniqueness) and use a more general set theoretical terminology instead. This terminology is the most general and even allows describing unconnected solution sets that can also not be handled, for example, by the Lie groups approach of Vasco (2007), and for which best-fitting oriented stochastic samplers will typically fail to find secondary extrema.…”