2-D reconstruction of boundaries with level set inversion of traveltimes

Zheglova, Polina; Farquharson, Colin G.; Hurich, Charles A.

doi:10.1093/gji/ggs035

Cited by 32 publications

(14 citation statements)

References 39 publications

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“…The level‐set method was originally developed by Osher and Sethian (1988) to determine the front propagation between two boundaries. This method has been successfully applied to non‐geophysical and geophysical applications such as in the two‐dimensional reconstruction of sharp boundaries in cross‐borehole tomography (Zheglova et al, 2013). It is important to note that we have not been able to come up with a precise rule on the number of subsets of fronts to be created and the number of flow lines to use.…”

Section: Theorymentioning

confidence: 99%

Estimation of Saturated Hydraulic Conductivity during Infiltration Test with the Aid of ERT and Level‐Set Method

Chou

Chouteau

Dubé

2016

Vadose Zone Journal

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Core Ideas A new technique for estimating the saturated hydraulic conductivity is developed. It uses a combination of hydrogeophysical and numerical methods. Electrical resistivity tomography and the instantaneous profile method are used. The technique is validated using simple and complex numerical hydrogeological models. Many hydrogeological and geophysical tools have been developed to determine subsoil properties, but they are often limited by sparse datasets and by the portability of the method from one site to another and often underestimate the complexity of the medium. We present a saturated hydraulic conductivity (Ks) estimation scheme, named the KES method, based on hydrogeophysical and numerical methods. The targeted medium of investigation is an unsaturated and heterogeneous soil. Estimation of Ks is accomplished by estimating the position of the wetting front and the distribution and velocity of flow lines during an infiltration test. Using numerical modeling, Ks is determined by minimizing the velocity difference between the measured flow lines and the modeled flow lines. Surface and buried electrodes are used as part of the electrical resistivity survey in determining the position of the wetting front. An instantaneous profile method is used to determine the water retention curve of the medium. The KES method has been tested and validated using data produced from simple and more complex geological models from published case studies. We obtained good reconstruction of the saturated hydraulic conductivity. We have found that the estimated value of Ks in log scale has a mean error <2.5%. Error increases along the boundaries of different hydrofacies.

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Section: Theorymentioning

confidence: 99%

Estimation of Saturated Hydraulic Conductivity during Infiltration Test with the Aid of ERT and Level‐Set Method

Chou

Chouteau

Dubé

2016

Vadose Zone Journal

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“…We work in an optimization framework, and overcome determination issues by restricting optimization over a prior or model subspace of functions and adding to the misfit functional a regularization functional. For piecewise constant slowness function we seek the interfaces between values defined by finite parameters [25,39], or with a finite basis [20] or infinite dimensional subspace [54,36]. One may use a probabilistic frame [51,29,13], where the prior is a space of probability distributions, but this is not within our scope.…”

Section: Introductionmentioning

confidence: 99%

“…The simplest choice α = 2 is problematic because s ∈ A ⊂ H 1 (Ω) does not give a well posed forward problem. A natural regularization for a binary recovery problem is to penalize the interfaces between constant values [45], (also [54,55]). If s ∈ BV (Ω, {a, b}), we can interpret (3.1) with α = 1 as the total variation of s, that is, the perimeter of the set {s = a}.…”

Section: Introductionmentioning

confidence: 99%

Binary recovery via phase field regularization for first-arrival traveltime tomography

Dunbar¹,

Elliott²

2019

Inverse Problems

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We propose a double obstacle phase field methodology for binary recovery of the slowness function of an Eikonal equation found in first traveltime tomography. We treat the inverse problem as an optimization problem with quadratic misfit functional added to a phase field relaxation of the perimeter penalization functional. Our approach yields solutions as we account for well posedness of the forward problem by choosing regular priors. We obtain a convergent finite difference and mixed finite element based discretization and a well defined descent scheme by accounting for the non-differentiability of the forward problem. We validate the phase field technique with a Γ -convergence result and numerically by conducting parameter studies for the scheme, and by applying it to a variety of test problems with different geometries, boundary conditions, and source -receiver locations.

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“…In terms of geophysical inverse problems, the level-set method has also found its wide applications, and the following citations are by no means complete. In [13], the level-set method was first applied to the gravity data; in [27] it was applied to identify uncertainties in the shape of geophysical objects using temperature measurements; in [41], [19] and [20] it was applied to travel-time tomography problems in different settings.…”

Section: Introductionmentioning

confidence: 99%

A multiple level set method for three-dimensional inversion of magnetic data

Qian

et al. 2017

SEG Technical Program Expanded Abstracts 2017

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We propose a multiple level set method for inverting three-dimensional magnetic data induced by magnetization only. To alleviate inherent non-uniqueness of the inverse magnetic problem, we assume that the subsurface geological structure consists of several uniform magnetic mass distributions surrounded by homogeneous non-magnetic background such as soil, where each magnetic mass distribution has a known constant susceptibility and is supported on an unknown sub-domain. This assumption enables us to reformulate the original inverse magnetic problem into a domain inverse problem for those unknown domains defining the supports of those magnetic mass distributions. Since each uniform mass distribution may take a variety of shapes, we use multiple level-set functions to parameterize these domains so that the domain inverse problem can be further reduced to an optimization problem for multiple level-set functions. To compute rapidly gradients of the nonlinear functional arising in the multiple-level-set formulation, we utilize the fact that the kernel function in the field-susceptibility relation decays rapidly off the diagonal so that matrix-vector multiplications for evaluating the gradients can be speeded up significantly. Numerical experiments are carried out to illustrate the effectiveness of the new method.

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2-D reconstruction of boundaries with level set inversion of traveltimes

Cited by 32 publications

References 39 publications

Estimation of Saturated Hydraulic Conductivity during Infiltration Test with the Aid of ERT and Level‐Set Method

Estimation of Saturated Hydraulic Conductivity during Infiltration Test with the Aid of ERT and Level‐Set Method

Binary recovery via phase field regularization for first-arrival traveltime tomography

A multiple level set method for three-dimensional inversion of magnetic data

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