Recovering the electrical conductivity of the soil via a linear integral model

Alba, Patricia Díaz de; Fermo, Luisa; Mee, Cornelis van der; Rodriguez, Giuseppe

doi:10.1016/j.cam.2018.11.034

Cited by 9 publications

(10 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The depth z and the height h are measured in meters, the electrical conductivity in Siemens per meter. The conditions for the existence and uniqueness of the solution of system (6.1) have been studied in [7], where three collocation methods were also proposed and compared. In [6], a preliminary version of the method presented in this paper was applied to the first equation of the model.…”

Section: Numerical Testsmentioning

confidence: 99%

“…Following [7], assuming the a priori information σ(z) ≤ β, for z > z 0 , we split each integral into the sum…”

Section: Numerical Testsmentioning

confidence: 99%

“…and assume that the values of the electrical conductivity at the endpoints of the integration interval are known, e.g., σ(0) = α and σ(z 0 ) = β. The boundary values can usually be approximated in applications; see [7]. By collocating equations ( 6.3) at the points h i , assuming n 1 = n 2 = n and h 1,i = h 2,i = h i , for i = 1, .…”

Section: Numerical Testsmentioning

confidence: 99%

“…Both estimation techniques appear to be effective. (z) for n = 20 by using an approach presented in [7] and the new proposed method. On the left noise level δ = 10 −4 , on the right δ = 10 −2 .…”

Section: Numerical Testsmentioning

confidence: 99%

“…The first equation describes the situation in which both coil axes are aligned vertically with respect to the ground level, while the second one corresponds to the horizontal orientation of the coils. This system has been studied in [7], under the assumption that the values of the unknown function at the boundaries are known, either on the basis of additional measurements or of known geophysical properties of the subsoil.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

de Alba,

Fermo,

Pes

et al. 2022

Preprint

View full text Add to dashboard Cite

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method to compute the minimal-norm solution in the presence of boundary constraints. The algorithm stems from the Riesz representation theorem and operates in a reproducing kernel Hilbert space. Since the resulting linear system is strongly ill-conditioned, we construct a regularization method depending on a discrete parameter. It is based on the expansion of the minimal-norm solution in terms of the singular functions of the integral operator defining the problem. Two estimation techniques are tested for the automatic determination of the regularization parameter, namely, the discrepancy principle and the L-curve method. Numerical results concerning two artificial test problems demonstrate the excellent performance of the proposed method. Finally, a particular model typical of geophysical applications, which reproduces the readings of a frequency domain electromagnetic induction device, is investigated. The results show that the new method is extremely effective when the sought solution is smooth, but gives significant information on the solution even for non-smooth solutions.

show abstract

Section: Numerical Testsmentioning

confidence: 99%

“…Following [7], assuming the a priori information σ(z) ≤ β, for z > z 0 , we split each integral into the sum…”

Section: Numerical Testsmentioning

confidence: 99%

Section: Numerical Testsmentioning

confidence: 99%

Section: Numerical Testsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

de Alba,

Fermo,

Pes

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

Truncated Minimal-Norm Gauss–Newton Method Applied to the Inversion of FDEM Data

Pes

2023

Lecture Notes in Computer Science

View full text Add to dashboard Cite

An Alternating Direction Multiplier Method for the Inversion of FDEM Data

Buccini,

de Alba,

Pes

2024

J Sci Comput

View full text Add to dashboard Cite

In this paper, we focus on the numerical solution of nonlinear inverse problems in applied geophysics. Our aim is to reconstruct the structure of the soil, i.e., either its electrical conductivity or the magnetic permeability distribution, by inverting frequency domain electromagnetic data. This is a very challenging task since the problem is nonlinear and severely ill-conditioned. To solve the nonlinear inverse problem, we propose an alternating direction multiplier method (ADMM), we prove its convergence, and propose an automated strategy to determine the parameters involved. Moreover, we present two heuristic variations of the ADMM that either improve the accuracy of the computed solutions or lower the computational cost. The effectiveness of the different proposed methods is illustrated through few numerical examples.

show abstract

Recovering the electrical conductivity of the soil via a linear integral model

Cited by 9 publications

References 28 publications

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Regularized minimal-norm solution of an overdetermined system of first kind integral equations

Truncated Minimal-Norm Gauss–Newton Method Applied to the Inversion of FDEM Data

An Alternating Direction Multiplier Method for the Inversion of FDEM Data

Contact Info

Product

Resources

About