Sviluppo di modelli numerici per la simulazione WP

Sviluppo di modelli numerici per la simulazione WP 2: Monitoraggio di inquinanti nel sottosuolo con inversione di dati Cristina Manzi, Ernesto Bonomi and Enrico Pieroni Environmental and Imaging Sciences, CRS 4 www. crs 4. it Environmental and Imaging Sciences 9/18/2020 1 MURST, Nov, 2003

WP 2: reconstruction and imaging Applied mathematical approach Developement of imaging strategies Inversion technique Reconstruction Environmental and Imaging Sciences 9/18/2020 2 MURST, Nov, 2003

Three strategies Linear inversion in frequency domain Non-linear inversion Multifrequency approach 2 D Time domain technique Environmental and Imaging Sciences 9/18/2020 3 MURST, Nov, 2003

Introduction (I) Geophysical EM surveys aim to provide information about conductivity of the Earth: n n Vadose zone characterization Ground water and salinity monitoring Detection of contaminants in soils and acquifers Detection of metallic debris From FEM measurements of the ground apparent electrical conductivity, the problem is to supply the conductivity profile of the subsurface Environmental and Imaging Sciences 9/18/2020 4 MURST, Nov, 2003

Introduction (II) Quantitative inference about subsurface conductivity is an ill-posed problem n n Least squares inverse problem Tikhonov regularization The conjugate gradient algorithm performs as a regularizing strategy without tuning of any parameters Environmental and Imaging Sciences 9/18/2020 5 MURST, Nov, 2003

EM 38 Instrument Fixed frequency: f=14. 6 k. Hz Fixed coil spacing: s=1 m Apparent conductivity (NB=s/d << 1): Hp primary field Hs secondary field d skin depth Horizontal and vertical configurations Environmental and Imaging Sciences 9/18/2020 6 MURST, Nov, 2003

EM 38 Linear Response Model Mc. Neill’s model for a stratified medium - s(z): conductivity at depth z - FH, V: sensitivity of the instrument Environmental and Imaging Sciences 9/18/2020 7 MURST, Nov, 2003

The Forward Model (II) Apparent conductivity [m. S/m] Height [m] Depth [m] Environmental and Imaging Sciences 9/18/2020 Conductivity profile [m. S/m] 8 MURST, Nov, 2003

The Inverse Model (II) Apparent conductivity [m. S/m] Height [m] Depth [m] Environmental and Imaging Sciences 9/18/2020 Conductivity profile [m. S/m] 9 MURST, Nov, 2003

Least Squares Problem Cost function: The minimum of e reached for the conductivity profile: Ill-conditioning Environmental and Imaging Sciences 9/18/2020 10 MURST, Nov, 2003

Tikhonov Regularization Enhance stability: trade-off between n and Ln: a discrete differential operator Solution Environmental and Imaging Sciences 9/18/2020 Condition number New least squares problem 11 a MURST, Nov, 2003

Tikhonov Regularization Enhance stability: trade-off between n and Ln: a discrete differential operator New least squares problem Solution Environmental and Imaging Sciences 9/18/2020 12 MURST, Nov, 2003

Conductivity [m. S/m] Inverse Problem Solution Depth [m][m] Environmental and Imaging Sciences 9/18/2020 13 MURST, Nov, 2003

The solver Constrain the optimal solution within the feasible set Projected conjugate gradient The problem is extremely ill-conditioned However: best solution for a=0, in the sense of proximity to the true conductivity profile Environmental and Imaging Sciences 9/18/2020 14 MURST, Nov, 2003

Projected Conjugate Gradient Projection strategy Convergence of the algorithm Conjugate gradient performs as a regularization Environmental and Imaging Sciences 9/18/2020 15 MURST, Nov, 2003

Conductivity [m. S/m] Borcher’s data set Depth [m] Environmental and Imaging Sciences 9/18/2020 16 MURST, Nov, 2003

A Field Data Example: the Poetto Beach Five soundings, every 10 m, along a profile orthogonal to the shore, starting 65 m before: n EM 38 height from 0 to 1. 5 m, with a 0. 1 m step, N=16 for each coil-mode configuration Near surface material: n n medium- to fine-grained sand (> 60% of quartz): 4 -5 m Sea water table depth, varying during the day: about 2 m Environmental and Imaging Sciences 9/18/2020 17 MURST, Nov, 2003

Apparent Conductivity [m. S/m] Vertical mode Environmental and Imaging Sciences 9/18/2020 Horizontal mode Top layer less conductive than the underlying ones 18 Height [m] MURST, Nov, 2003

Subsurface Conductivity [m. S/m] M=100 layers, a=0 Depth [m] Sand+air with a 30 -40% porosity: low conductivity Environmental and Imaging Sciences 9/18/2020 Sand fully saturated by salt water: high conductivity 19 MURST, Nov, 2003

Multifrequency analysis Non-linear inversion of the magneto-telluric equation n Forward problem: n Adjoint problem: n Minimization: projected conjugate gradient Environmental and Imaging Sciences 9/18/2020 20 MURST, Nov, 2003

Two strategies Construction of intermediate solution supplied by the conjugate gradient • Average Solution (AS) • Average Gradient (AG) Cost function: Global gradient: Environmental and Imaging Sciences 9/18/2020 21 MURST, Nov, 2003

Conductivity [m. S/m] An example Depth [m] Environmental and Imaging Sciences 9/18/2020 22 MURST, Nov, 2003

TDEM technique Maxwell’s equations in time domain Finite element discretization in spatial domain Crank-Nicolson scheme for time discretization Secondary electric field Environmental and Imaging Sciences 9/18/2020 23 MURST, Nov, 2003

TDEM technique Spatial discretization Temporal discretization • Mitsuhata boundary conditions Environmental and Imaging Sciences 9/18/2020 24 MURST, Nov, 2003

TDEM - Numerical examples Two layered soil with different resistivity Time discretization: Environmental and Imaging Sciences 9/18/2020 25 MURST, Nov, 2003

TDEM – Numerical examples Environmental and Imaging Sciences 9/18/2020 26 MURST, Nov, 2003

TDEM – Numerical examples Variation of the secondary electric field 15 m above the current line on the surface In late time the slope of the curve is: Environmental and Imaging Sciences 9/18/2020 27 MURST, Nov, 2003

Conclusions I: Linear model Regularization: Tikhonov or Conjugate Gradient Ill-conditioned problem Stability Projected Conjugate Gradient Experimental data provide credibility to our results on the EM 38 linear inversion strategy Environmental and Imaging Sciences 9/18/2020 28 MURST, Nov, 2003

Conclusions II: Non Linear model Local minima Constraints projection Data and instruments? Ottimization strategy… Environmental and Imaging Sciences 9/18/2020 29 MURST, Nov, 2003

Conclusions III In the multi-frequency model, the inversion combines an iterative scheme implementing a constrained non-linear conjugate gradient n The AS method gives a good accuracy for shallow and deeper layers while the AG method performs correctly only in the near zone surface TDEM direct algorithm provides a reliable reconstruction of the secondary electric field for a 2 D medium excited by a infinite line source TDEM inversion Environmental and Imaging Sciences 9/18/2020 30 MURST, Nov, 2003

COLLABORATION WITH: • Gian Piero Deidda, Departement of Territorial Engineering, UNICA • Brian T. Borchers, Department of Mathematics, New Mexico Tech, Socorro • Eiichi Arai, Metal Mining Agency of Japan • Yuji Mitsuhata, National Institute of Advanced Industrial Science and Technology, Japan Environmental and Imaging Sciences 9/18/2020 31 MURST, Nov, 2003