Joint inversion of crosshole radar and seismic traveltimes acquired at the South Oyster BacterialTransport Site Page: 3 of 33
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natural extension to two- and three-dimensional applications has been to assume that the earth
can be divided into sub-volumes of uniform properties with geometries that are common for
all physical properties in the inversion (e.g., Hyndman and Gorelick, 1996; Musil et al.,
2003). Such approaches are certainly useful, but physical properties can vary gradually in
space and not all data are necessarily sensitive to the same changes in lithology and state
variables. Furthermore, the zonations must be updated continuously, making the inversions
In Occam's inversion (Constable et al., 1987), fine model discretizations are used and
the inverse problem is regularized by minimizing, for example, model roughness with the
constraint that the simulated model response is close to a given target data misfit. Haber and
Oldenburg (1997) introduced a joint inversion scheme to find models that are structurally
similar, in the sense that spatial changes in models occur at the same location. This scheme is
applicable to over-parameterized two- and three-dimensional models and it is essentially
based on minimizing the squared difference of a weighted Laplacian of the two models.
Gallardo and Meju (2003) further developed the framework of the structural approach to
joint inversion proposed by Haber and Oldenburg (1997) by defining the cross-gradients
function t(x,y,z) as
t (x, y,z) =VmA (x, y,z) xVmB (xyz, (1)
where VmA (x, y, z) and VmB (x, y, z) are the gradients of models mA and mB at location x, y,
and z, and x indicates the cross-product. By forcing the discretized cross-gradients function to
be close to zero at each location during the inversion process, either the gradients of the two
resulting models will be parallel or anti-parallel to each other or one or both of the models
does not change. The boundaries of the resulting models have the same orientation, thus
facilitating geological interpretations. An advantage compared to the work of Haber and
Oldenburg (1997) is that constraints based on the cross-gradients function do not focus on the
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Linde, Niklas; Tryggvason, Ari; Peterson, John & Hubbard, Susan. Joint inversion of crosshole radar and seismic traveltimes acquired at the South Oyster BacterialTransport Site, article, April 15, 2008; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc901262/m1/3/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.