Intersections, ideals, and inversion

Water Resources Research

Karasaki

2006

[1] An asymptotic expression for transient head variations, valid at low-frequencies, forms the basis for an efficient scheme for estimating hydraulic conductivity. The critical computational step is equivalent to solving the governing equation for steady state head. Thus, model parameter sensitivities, relating changes in head to changes in hydraulic conductivity, of the fully transient problem can be computed with the equivalent of four steady state head computations. A comparison of model parameter sensitivities computed using the low-frequency asymptotic approach and sensitivities computed using a purely numerical approach indicates good agreement. An inversion of synthetic hydraulic tomography data indicates that it is possible to estimate overall permeability variations using the technique. In an actual application to truncated crosswell pressure tests from the Raymond field site, we image two high permeability fracture zones, in agreement with a conceptual model of the region. The location of the two fracture zones correlates with the position of transmissive fractures, as measured by borehole conductivity logs.

Section: Applicationsmentioning

confidence: 99%

Interpretation and inversion of low‐frequency head observations

Water Resources Research

Karasaki

2006

“…In Everett (1996) and Vasco (1998), governing equations such as (4) were discretized using a finite difference methodology. In Appendix A the method of weighted residuals, a basis for the finite element method, is used to approximate (4) by a system of algebraic equations.…”

Section: Methodsmentioning

confidence: 99%

“…(12). In Vasco (1998) the penalized least‐squares approach for differential equations such as (3) is discussed in more detail. One writes eq.…”

Section: Methodsmentioning

confidence: 99%

“…For example, a negative conductivity may result, or the conductivity or ¢eld components might exceed some prior bound. This was the case in Vasco (1998), where only one of 25 solutions was found to be physically plausible. The solutions to the inverse problem are the roots of the algebraic equations (44).…”

Section: The Battle Mountain Magnetotelluric Experimentsmentioning

confidence: 99%

“…Restricting x to observation points on the surface x i and noting that the ¢rst term on the right is the predicted surface ¢eld U 0 (x i ), I may write (Vasco 1998). As with di¡erential equations, discretization is an e¡ective tool when solving integral equations and evaluating functionals (Noble 1964;Green 1969;Linz 1985;Hackbusch 1995).…”

Section: Integral Equations and Functionalsmentioning

confidence: 99%

See 2 more Smart Citations

An algebraic formulation of geophysical inverse problems

Geophysical Journal International

2000

Many geophysical inverse problems derive from governing partial differential equations with unknown coefficients. Alternatively, inverse problems often arise from integral equations associated with a Green’s function solution to a governing differential equation. In their discrete form such equations reduce to systems of polynomial equations, known as algebraic equations. Using techniques from computational algebra one can address questions of the existence of solutions to such equations as well as the uniqueness of the solutions. The techniques are enumerative and exhaustive, requiring a finite number of computer operations. For example, calculating a bound to the total number of solutions reduces to computing the dimension of a linear vector space. The solution set itself may be constructed through the solution of an eigenvalue problem. The techniques are applied to a set of synthetic magnetotelluric values generated by conductivity variations within a layer. We find that the estimation of the conductivity and the electric field in the subsurface, based upon single‐frequency magnetotelluric field values, is equivalent to a linear inverse problem. The techniques are also illustrated by an application to a magnetotelluric data set gathered at Battle Mountain, Nevada. Surface observations of the electric (Ey) and magnetic (Hx) fields are used to construct a model of subsurface electrical structure. Using techniques for algebraic equations it is shown that solutions exist, and that the set of solutions is finite. The total number of solutions is bounded above at 134 217 728. A numerical solution of the algebraic equations generates a conductivity structure in accordance with the current geological model for the area.

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

Geophysical Journal International

2007

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 .