AbstractâThis paper presents a simple and fast algorithm to analyze wideband electromagnetic induction (WEMI) data for subsurface targets. A well known four ...
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0) is: −1+ (ωτ )c cπ . tan m(ω) = −2 tan−1 c 4 1 + (ωτ )
(5)
(6)
EF = (7) for
(9)
The lookup table error EL is defined as follows: (10) EL = Eˆ (cˆ,τˆ) where cˆ,τˆ = arg min Eˆ (c,τ ) . c,τ The amplitude A was found by substituting the estimates τ and c in (6) and finding a least squares estimate: Nf
k
)
(11)
k =1
cˆπ cˆ 3(ωkτˆ ) sin 2 ∑ k =1 Nf
∑ Z (ω )
.
(13)
2
k =1
(8)
ω
∑ Q(ω
Nf
2
k
absolute error:
Aˆ =
cˆ ˆ sˆ + ( jωkτˆ ) − 2 Z ( ) A − ω ∑ k ( jωkτˆ)cˆ + 1 k =1 Nf
The range of m( ω) is given by its value at the extrema of ωτ : cπ cπ . and m (ω ) ωτ = ∞ = − m (ω ) ωτ = 0 = 2 2 These equations show that the gradient angle varies between cπ/2 and –cπ/2 which is adequate to model all angles when the value of c ∈ [0, 2]. They also show that the three-parameter model developed in [1], where c is restricted to be equal to 0.5, can only characterize angles between -π/4 and π/4. From our data, we found that to be inadequate to model most landmines. Since (8) involves only two variables, a lookup table was created for m(ω) for a range of values of τ and c. The values of τ and c were found by matching m( ω) (numerically computed using (7)) with the mˆ (ω , c,τ ) table values for least mean
Eˆ (c,τ ) = ∑ m(ω ) − mˆ (ω , c,τ ) .
cˆπ cˆ 31 + (ωkτˆ ) cos (12) 2 1 I (ωk ) sˆ = −1− . ∑ ˆ N f k =1 cˆπ cˆ 2 cˆ A 1 + (ωkτˆ ) + 2(ωkτˆ ) cos 2 The goodness of fit for this method can be measured as the normalized error between the actual data and the fit: Nf
cˆπ 2 cˆ cˆ 1 + (ω kτˆ ) + 2(ωkτˆ ) cos 2
where τˆ and cˆ are lookup table estimates, and Nf is the number of measured frequencies. Finally, the shift s was found by substituting the estimates of A, τ and c in (5):
Fig. 1. m vs. ωτ and c. It shows a discontinuity in m when c = 2 and ωτ = 1.
D. Stability of Lookup Table Method All the parameter estimates depend on the accuracy of the gradient and on the accuracy of lookup table estimates of τ and c. First it is necessary to see how a small error in the parameter estimate in the (τ, c) space due to the finite size of the lookup table affects the angle estimate. Fig. 1 shows the dependence of m on τ and c. It shows two regions of interest, one being c = 0, and another c = 2. Fig. 2 shows that near regions where ωτ = 1, there is a large change in m for small changes in τ and c due to the fact that the argument in the right hand side of (8) becomes zero. The stability analysis is done by studying the effect of perturbation in τ and c. Next, it is necessary to analyze the robustness of the estimated parameters with respect to noise in the data and in the gradient estimates. Since the noise in the parameters is non-linearly linked to the noise in the data, it is difficult to derive expressions linking the noise in them. Therefore, the stability of the method with respect to noise in the data will be analyzed in the context of classifier design.
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