Abstract

The sensing and detection of dense non-aqueous phase liquid (DNAPL) contaminations in soil has significant geo-environmental benefits; and it is challenging because the background media is uncertain, wave characteristics of media may be hard to determine, and wave scattering of non-ideal volumes in non-uniform backgrounds is difficult. Sensing the subsurface volume between boreholes using cross-well radar (CWR) is less expensive and saves effort, but has not been investigated satisfactorily. In this work, we developed an analytical model to approximate CWR sensing in infinite half-space lossy media in frequency domain. Half-space dyadic Green's function due to a vertical polarized dipole source is introduced. Integration for angles gets far into evanescent range. Born approximation is employed as a linear model for a shape-based inversion that is developed to localize the object assuming its contrast to the lossy background is a priori information. This forward model is validated via CWR experiment. Soil parameters (relative dielectric constant and loss tangent) variance with frequency is represented by a quadratic polynomial. Calibration for soil parameters is conducted with CWR data using an iterative low-order parameterized optimization technique involving both magnitude and phase information. The validated forward model can predict for CWR sensing experiments in a broad frequency range very well. Localization and reconstruction of the object is performed as a non-linear least square optimization problem by minimizing a cost function that calculates the misfit between the predicted numerical simulation and experimental observation. The proposed inversion method is validated by numerical experiments, and it gives promising preliminary results.

State of the Art

There are different detection techniques implemented in the field, such as Direct Push Probe Technologies (DPT), In-Situ Tracers (IST), Geophysical Methods (GM). GM methods are the most non-invasive ones to locate subsurface DNAPLs; avoiding the risk of additional vertical migration of pooled DNAPL. Of course, GPR is the least non-invasive, but has depth limitation. Thus, CWR has been chosen to study in this research. The prospective algorithms in solving the pool characterization problem are based upon multi-scale image formation and shape-based inversion methods for localizing parameterized pollution pools.

Forward Modeling, Validation, Calibration and Shape-based Inversion for Geophysics Problem

He (Sophia) Zhan, Carey M. Rappaport, Mohammad Farid, Akram N. Alshawabkeh, Eric L. Miller, Harold RaemerDepartment of Electrical Engineering, Northeastern University, MA 02115

This work was supported in part by CenSSIS, the Center for Subsurface Sensing and Imaging systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC- 9986821)

Value added to CenSSIS

R1

FundamentalScienceFundamentalScience

ValidatingTestBEDsValidatingTestBEDs

L1L1

L2L2

L3L3

R3

S1 S4S3 S5Bio-Med Enviro-Civil

S2

R2

Develop and verify innovative computational tools for real time efficient, non-invasive, cost-effective and reliable monitoring, delineation and characterization of DNAPLs using Ultra-Wideband Cross-Well Radar.

• Dense non-aqueous phase liquid (DNAPL) contaminants 0 (2.6 – 0.001j)– Object is wide and thin (pool)– Side width to thickness ration is

bigger than 10 typically– relative dielectric constant and

loss tangent) are frequency dependent

• Saturated sandy soil (lossy medium) 0 ( 21 – 0.08j)

• Reconstruct and detect the location and the size of the anomaly

Problem Description

Objective

Half-space Scattered Field with Born Approximation

The scattered field as a function of source location, receiver location and source frequency, involves a sensor transfer function (dyadic Green’s function and background field ) and the perturbation of object function.

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• Electric field for 3D electromagnetic problems in frequency domain – vector Helmholtz equation

– Assumption: object and homogeneous background are additive

• Scattered E-field with first order Born Approximation

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kwhere is the wave number; is the complex permittivity and is the complex permeability; is the source.

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•Plane wave decomposition•Fresnel reflection for each plane wave at interface•Integration for angles far into evanescent range

Half-Space Lossy Green’s Function Model

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Validation: Air/Sand Half-Space Model vs. Finite Difference Freq. Domain (FDFD)

Pilot-scale (1/100) of CWR imaging•8 transmitters / 8 receivers•Positioned on a circle of radius 6”

Antenna specifically developed for CWR•Good wideband (0.5 ~ 2.2GHz) coupling to dry/wet sand•Fabricated for repeated insertion in PVC boreholes

Cross-Well Radar Sensing Setup

Sensor Independent Transformation: Channel Transfer Function (CTF)• In cross-well-radar experiment, coupling between monopole antenna and soil

varies with frequency • In forward modeling, wave propagation that is independent of the sensor is

studied• CTF is introduced to translate the real experimental data to the form that is

compatible to the simulation by forward model

The three cascaded two-port junctions with unknowns is matching to the one two-port junction obtained via experiments. Since the soil is lossy, the reflection coefficient at soil junction can be neglected. The magnitude of CTF is solved as:

S21 = Transmission measurements S11 = Reflection measurements

0(f) = phase offsetThe assumption in solving the CTF could affect the phase of the CTF, a progression analysis is conducted to find the appropriate offset amount to the phase.

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Source depth is 20 below interface in z-direction and centered in xy plane; observation plane is 20 apart from source, where at f=1GHz;The relative dielectric of the lossy media is

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Calibration via CWR Experiments

• Analysis– Loss tangent variance will only affect the magnitude of the electric field– Relative dielectric constant change affects both the magnitude and the

phase• Assumptions

– Quadratic polynomial as a function of frequency is sufficient to describe the parameter variances with frequency

• Approach– Minimize a cost function in the sense of nonlinear least square error

– Obtain optimal polynomial coefficients using Lavenberg-Marquardt Method

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Calibration results for soil parameters

Relative dielectric constant Loss target

Experiment vs. Model

Network analyzer measurementsHalf-Space Lossy Green’s Function, initial soil calibrationOptimal soil selection

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Shape-based Inversion

• frequency = 1.5GHz ; • Additive Gaussian noise SNR = 20dB• Region of interest is 9cm x 9cm x 0.75cm (15x15x15) x = y 6mm ; z 0.5mm ’sand = 20 – 0.14j • ’obj = 2.6 – 0.001j

• Linear array of T/R -- 7 vertical locations and 4 corner positioning• Levenberg-Marquardt method

Numerical Experiment

• Find a geometry function such that the voxels have value 1 inside the ellipsoid, 0 elsewhere [8]

• Ellipsoid formulation– Entries of diagonal matrix D are the inverse of the lengths of the semi-axes

(lx,ly and lz)

• Geometry function

• Use Levenberg-Marquardt Method• Compute Jacobian matrix analytically

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Ground truth:[xo,yo,zo, lx, ly, lz] = [4.764452 4.764452 0.400000 4.168896 4.1688960 0.2500000]cm

Estimated result:[xo,yo,zo, lx, ly, lz] = [4.764460 4.764484 0.400014 4.168896 4.1688959 0.2500099]cm

1) M. L. Brewster and A. P. Annan, “GPR Monitoring of a Controlled NAPLRelease: 200 MHZ Radar”, Geophysics, Vol. 59, No. 8, August 1994

2) L. Tsang, J.A. Kong and R. R. Shin, “Theory of Microwave Remote Sensing”, JOHN WILEY & SONS, 1985

3) C. Balanis, “Antenna Theory: Analysis and Design”, Harper & Row, 19824) J. Hipp, “Soil Electromagnetic Parameters as Functions of Frequency, Soil Density, and

Soil Moisture”, Proc IEEE, Vol. 62, pp. 98-103, Jan. 1974.5) C. R. Vogel, “Computational Methods for Inverse Problems”,SIAM,20026) R. E. Collin, “Foundations for Microwave Engineering”,2nd ed., 19927) Misha E. Kilmer, Eric L. Miller, Alethea Barbaro and David Boas, “Three-dimensional

shped-based imaging of aborpton perturbation for diffuse optical tomography”, Applied Optics (2003), 42, 3129 – 3144

8) Basak, U. Karbeyaz, “Reduced Complexity Geometry-Based Born Inversion for Frequency Domain Ultrasonic Monitoring of Cancer Treatment”, Ph.D thesis, Sept., 2005

References

Future Works

• Segmentation of region of interest with non-uniform voxel to better match the shape of the object while preserving good resolution

• Experiment vs. model comparison with DNAPL• Inversion using multi-frequency information• Inversion algorithm validation with CWR data

Initial guess volumeReconstructed volume

1) Zhan, H., Farid, M., Rappaport, C. M and Alshawabkeh, A. N., “ Born Approximation Modeling of Lossy Soil Half-Spaces for Cross-Well Radar Sensing of Contaminants with Low-Order Parameter Optimization”, in submission

2) Farid, M. , Zhan, H., Alshawabkeh, A. N, and Rappaport, C. M., “Cross Well Radar : II – Comparison and Validation of Experimentation and Modeling Validation using Channel Transfer Function”, in submission

3) Zhan, H., Rappaport, C., Farid, M., Akram, A. and Raemer, H., “Lossy Halfspace Born Approximation Modeling of Electromagnetic Wave Source and Scattering in Soil by Cross Well Radar”, ASCE Geo-frontiers, Austin, TX, Jan. 2005, electronic CD

4) Farid, M., Akram, A., Zhan, H. and Rappaport, C., “Challenges and Validation of Cross Tomography Experimentation for Inverse Scattering Problems in Soil”, ASCE Geo-frontiers, Austin, TX, Jan. 2005, electronic CD

5) Farid, M. et al. (2003), "Experimental DNAPL Detection Using Cross-Well Radar", SAGEEP, San Antonio, TX. USA, Electronic Proceeding

6) Farid, M. et al. (2003), "Modeling Borehole Dipole Antenna Patterns for Cross-Well Radar DNAPL Imaging", Soil-Rock Conference, MIT, Boston, MA. USA, proceeding, pp. 261-267

7) Farid M. et al. (2002), "DNAPL Detection Using Cross-Well Radar", Environmental Geotechnics, 4th ICEG, pp. 465-470

8) M. Farid, A. Alshawabkeh, C. Rappaport (2003) “Experimental DNAPL Detection Using Cross-Well Radar", SAGEEP, St. Antonio, TX.

Publications Acknowledging NSF Support