Application of Bayesian Inversion for Scatterer Shape from EMI Data ... electromagnetic induction (EMI) responses at different locations, in terms of scattered.
2003 IEEE AP-S Int'l Symp. & USNC/CNC/URSI Nat'l Radio Sci. Mtg, Columbus, OH, June 22-27, 2003
Application of Bayesian Inversion for Scatterer Shape from EMI Data K. Sun*2, K. O’Neill1, Lanbo Liu1, F. Shubitidze2, and I. Shamatava2 1
Engineer Research and Development Center US Army Corps of Engineers 72 Lyme Road, Hanover, NH 03755
2
Thayer School of Engineering at Dartmouth College Hanover NH 03755
Abstract Bayesian inversion approaches may be useful for inferring metallic scatterer shapes, and thereby assist in discriminating buried unexploded ordnance (UXO). The fundamental feature of Bayesian inversion is its attempt at rational incorporation of prior information in the inference algorithm. In UXO detection and classification, the model is a set of parameters corresponding to a particular object in a particular disposition. Prior information about the target sought and the randomness of noise and clutter from different sources warrant the application of a Bayesian approach. Broadband electromagnetic induction (EMI) responses at different locations, in terms of scattered magnetic field components in-phase and out-of-phase with the transmitted primary field, form the data vector. Here a fast forward model, in which we successfully represent an steel cylinder with a spheroid [2,3,4], is exploited for inversion computations. The validated model produces synthetic data, for which Bayesian inversion is compared to simple least squares (SLS), i.e. without weighting or damping. The Bayesian approach can provide more accurate results, if we can provide reasonable prior information. This work was sponsored in part by the Strategic Environmental Research and Development Program and US Army CoE ERDC BT25 program. I. Introduction Cleaning up of UXO is extremely expensive, partly because of the difficulty of distinguishing UXO from metallic targets of no interest. EMI sensing has been shown to be a promising technique to address this problem. Here we will discuss an approach for inferring target parameters from EMI data. A fast forward model applied in the inversion is based on new analytical solutions for scattering from spheroids [2-4]. In a certain distance range, a spheroid may be a better representative of the object than a very small number of infinitesimal dipoles, a simple model that is commonly employed [e.g.1]. The Bayesian inversion approach has gained much popularity in recent years with increasing recognition and appreciation of the stochastic nature of a wide range of inverse problems [4,5]. In Bayesian inversion prior information is characterized probabilistically and incorporated directly it into the inversion algorithm. Here we will first test our forward model against measured data, then the validated model is used to produce synthetic data for testing our Bayesian approach against SLS. II. Bayesian algorithm for inferring parameters. Nine parameters m comprise the model vector, which identifies the spheroid (and also for the target it represents), specifically m = [x0,y0,z0,θ0,φ0,a,b,σ,µr], where the point (x0,y0,z0) is the center of target in global coordinates and ,θ0 and φ0 are its polar and azimuthal orientation angles, respectively. In coordinates aligned with its principal axes,
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the spheroid can be described by function x 2 a 2 + z 2 b 2 = 1 . σ is the electrical conductivity (S/m) and µ r is the relative magnetic permeability of the target. To avoid
the difficulty of inferring σ and µ r simultaneously [4], we usually assume one of them is known, so that only 8 parameters are to be inferred. The measured data d = ( d1 d 2 ! , d N )
T
can be written in a general form as [7]
d = f (m ) + n , where f (m ) is the ‘pure’ EMI signal from the target. Assume the noise
n = (n n ! , n )
T
1
(σ
N
2
is additive and has normal distribution with standard deviation
σ 2 ! , σ N ) . Then for a set of data with N measurements, the probability of any data T
1
vector, given a particular set of model parameters m, is [8]
1
P(d|m) =
(2π ) N / 2 Cd
1/ 2
1 exp − (d − f (m)) ⋅ Cd−1 ⋅ (d − f (m)) 2
(1)
where the covariance matrix is Cd = diag[σ 1 , σ 2 ,..., σ N ] . Bayes’ Theorem for Inversion can be expressed as [7] 2
P(m | d) =
P(d | m) P(m) P(d )
2
2
(2)
where P (m | d) is the pdf of the model parameters m posterior to the data d; P (m) is the prior probability of the model vector m, available through our judgment or experience; P(d) is the probability of any data vector d, which is generally estimated from field data. We seek the set of m that maximizes P(m | d) , assuming the form (1).
II. Results and Discussion To investigate the possibility of representing a general object with a spheroid, we choose a steel cylinder (diameter d = 4cm, length L = 24cm) as an example. The EMI data were measured over a 5 by 5 grid on a surface above the target, and the total size of the grid is 50cm by 50cm. A global coordinate system (x, y, z) was established on the grid surface, with origin at the center and z-axis perpendicular to the surface. The target center is at (x0, y0, z0) = (2cm, -11.5cm, -20cm) in the global system. The orientation corresponds to θ0 = 450, φ0 = 900 relative to the x axis. At each grid point the scattered inphase and phase-quadrature magnetic field components (relative to the primary field) were recorded with a GEM-3 sensor for 17 frequencies. The spheroid parameters are estimated by SLS, i.e. by minimizing Φ=
25
17
∑ ∑
nobsv =1 Nf =1
2
H data − H m , where H data is the measured data and H m indicates values
from the model. The minimization gives the inferred parameters as a=3.76cm, b=16.63cm, x0=2.2cm, y0=-12.27cm, z0=-24.1cm, θ0=42.50, φ0=85.570, σ= 4.4e6 s/m, µr=112. Figure 1 shows the model (solid lines) and measured H fields (markers)
over 17 frequencies, at points (0,0,0) and (0,-20cm,0 ) in global coordinates. Clearly the modeled data fit the measurements very well.
f r e q u e n c y r e s p o n s e a t p o in t ( 0 , 0 )
f r e q u e n c y r e s p o n s e a t p o in t ( 0 , - 2 0 c m ) 400
4 000
Q u a d r a tu r e
2 000
R e ( H ) & I m (H )
R e (H ) & Im (H )
200 0 -2 0 0 -4 0 0
Q u a d ra tu re
0 -2 0 0 0 -4 0 0 0 -6 0 0 0
In p h a s e
In p h a s e -6 0 0 -8 0 0 10
-8 0 0 0 100
1 000
10
4
10
5
-1 1 0
4
10
100
1 000
10
4
10
5
f(H z )
f(H z )
Figure 1. Scattered fields vs. frequency at two observation points. To demonstrate a Bayesian approach, we calculate the EMI scattered field by numerical simulation, adding in artificial noise to synthesize noisy datum, and then infer parameters of the representing spheroid. The observation points and frequencies are the same as for the cylinder in last section, but the target is a spheroid with a = 3cm , b = 18cm , x0 = 0 , y0 = 0 , z0 = −30cm , θ 0 = 450 , σ = 4 ×106 s / m , µ r = 100 . The noise has normal distribution with mean µ ni = 0.1d i and standard deviation σ ni = 0.25di . Where
di is the ith component of data d. To simplify the problem, we assume the location, orientation, conductivity and magnetic permeability are known, so that only a and b, two of the most important parameters for UXO discrimination, are to be inferred. The SLS approach, without accounting for noise, and the Bayesian approach were employed on 100 sets of synthetic data and results were compared. The standard deviation of the data is known, here the same as that of the added noise, i.e. σ i = 0.25di + ε , with ni = 0.. The small threshold ε =1x10-6 is added to provide a noise floor relative to small data magnitudes [1] and so that the algorithm remains non-singular. In addition, we assume we know some prior information in terms of pdf's for a and b, i.e. we are looking for a specific target type:
1 b − b 2 1 a − a 2 1 1 0 0 exp − exp − P( a) = , P (b ) = 1/ 2 1/ 2 (2π ) σ a (2π ) σ b 2 σ a 2 σ b
(3)
where a0 = 3.2 cm, b0 = 17.5 cm, σ a = 0.1 cm, σ b = 0.1 cm . Substituting into (1) and (2), drop off the terms that are not related to a and b, take logarithm, we will obtain the objective function to be minimized as: 2
a − a0 b − b0 Φ(m) = (d − f (m) − µ m ) C (d − f (m) − µ m ) + + σa σb T
−1 d
2
(4)
Results from both algorithms are shown in Figure 2. Clearly, the Bayesian approach gives more accurate results in this example.
µ 2 0
a
le a s t s q u a r e , = 3 .0 9 4 c m , σ = 0 .0 8 c m a
µ 1 6
b
le a s t s q u a re , = 1 8 .4 9 c m , σ = 0 .6 c m b
1 4 1 2
C ou nt
1 5
1 0 8
1 0
6 4
5
2 0 2 .7 9
2 .9 1
3 .0 3
3 .1 5
3 .2 7
0 1 6 .8
B a y e s ia n , µ 1 0 0
a
= 3 .0 0 5 c m , σ
a
1 8 .4
1 9 .2
2 0
B a y e s ia n ,
= 0 .0 0 5 8 c m
µ 6 0
b
= 1 8 .2 6 7 c m , σ
b
= 0 .0 5 2 6 c m
5 0
8 0
C o u n t
1 7 .6
4 0 6 0 3 0 4 0 2 0 2 0 0 2 .7 9
1 0 2 .9 1
3 .0 3
R a n g e o f
3 .1 5
a (c m )
3 .2 7
0 1 6 .8
1 7 .6
1 8 .4
R a n g e o f
1 9 .2
2 0
b (c m )
Figure 2 Histogram of inferred a and b values from SLS and a Bayesian approach. Conclusions A fast forward model was introduced and tested for analytical forward modeling of the EMI scattered field from metallic targets, using a spheroid. A Bayesian shape inversion approach, utilizing statistical prior information, proved superior to simple least squares in elementary test cases with added Gaussian noise. This suggests potential for application to UXO detection and discrimination.
References 1.
Pasion, L.R., and Oldenburg, D.W., A Discrimination Algorithm for UXO Using Time Domain Electromagnetics: J. Engg & Envir. Geophys, 20 (2), 91-102,2001. 2. C. O. Ao, H. Braunisch, K. O’Neill and J. A. Kong Quasi magnetostatic solution for a conducting and permeable spheroid with arbitrary excitation, IEEE Trans, Geosci. Rem. Sens., Vol. 40,, 2002. 3. I. Shamatava, K. O’Neill, F. Shubitidze, K. Sun and C.O. Ao, Evaluation of Approximate Analytical Solutions for EMI Scattering from Finite Objects of Different Shapes and Properties," trans. IGARSS 2002, pp1550-1552 4. K. Sun, K O’Neill, I. Shamatava, F. Shubitidze, Application of Prolate Spheroid Solutions in Simulation of EMI Scattering with Realistic Sensors and Objects, ACES conference Mar 2428, 2003 Monterey, CA, to appear 5. L. Collins, P. Gao, D. Schofield, J. P. Moulton , L. C. Makowsky, D. M. Reidy, and R. C. Weaver, A Statistical Approach to Landmine Detection Using Broadband Electromagnetic Induction Data. IEEE Trans. Geosciences and Remote, 40(4), pp 950-962, 2002. 6. K. Sun, K. O’Neill, F. Shubitidze, I. Shamatava, K.D. Paulsen, “Theoretical analysis of TSA formulation and its domain of validity,” submitted for publication. 7. J. A. Scales and L. Tenorio, Tutorial: Prior information and uncertainty in inverse problems, Geophysics, vol.66, pp389-397, 2001. 8. T. J. Ulrych, M. D. Sacchi, and A. Woodbury, A Bayes tour of Inversion: A tutorial, Geophysics, 66, pp55-69, 2001.
