estimation method of the radius, depth and direction of...
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Estimation Method of the Radius, Depth and Direction of Buried Pipes with Ground Penetrating Radar
Yoshihiko NOMURA1), Yoichi NAGANUMA1), Hirofumi KOTAKI1), Norihiko KATO1), Yoshikazu
SUDO2) 1)Graduate School of Engineering, Mie University, 514-8507, Tsu , JAPAN
2) IS Engineering Co., Ltd. Chiba, JAPAN
Abstract: - By using a nonlinear least-squares method to the propagation time data obtained by GPR (Ground Penetration Radar), we proposed an estimation method of the geometric conditions of buried pipes. They are estimated so that the geometrical model-calculated propagation times are fit to the observed ones at all antenna positions. The calculated propagation time is assumed to be given by the shortest path between the antenna and the buried pipe. Comparing conventional method to proposed method of experimental results, we confirmed that the proposed method is effective enough to estimate the geometrical conditions of a buried pipe. Key-Words: GPR, nonlinear least-squares method, buried pipe, direction, position, diameter, measurement
1 Introduction Currently, GPRs (Ground Penetration Radar systems) are widely used to detect underground buried objects because they can nondestructively visualize underground situation. However, observed GPR images known as B-Scope images are a kind of point spread function (PSF)-filtered images, and, are blurred. Such blurred images have poor resolution due to directional characteristics. In addition to that, they suffer from noises due to the non-uniformity of propagation medium.
Many techniques have been developed to encounter the degraded images and to reveal the subsurface structures. The techniques are classified into two primary categories. One is the nonparametric imaging algorithm such as SEABED [1, 2], the envelope method [3], and RPM (Range Points Migration) method [4]. The other is the parametric estimation method. For example, buried objects are modeled by some parameters, and are recognized and measured in the manner of parameter estimation. That is, parametric inverse transformation techniques have been proposed to estimate the relative permittivity of the soil medium and pipe locations using propagation time information from antenna units to buried pipes [5], [6], and to buried plates [7]. Reference [8] evaluated the estimate accuracy of the inverse transformation technique and examined the ill effects on the
estimated values due to propagation time measurement error and relative permittivity assumption error.
In this paper, we propose a novel synthetic aperture method being classified into the latter parametric method. The unknown parameters are the radius, depth and direction of buried pipes. As for the regression equations, propagation time is ingeniously modeled in the form of non-linear equations.
2 Principle of Ground Penetrating Radar (GPR) The antenna unit is composed of an emitter and a receiver (see Fig. 1 and Fig. 2). The emitter radiates an electromagnetic wave pulse toward underground. The pulse length is about 1.0 nsec, and this antenna’s center frequency is 700MHz. The electromagnetic wave pulse propagates in the soil medium and is reflected by the boundary between materials having different permittivities. The transmitting time, yT is given by
0
22
c
l
c
lyT
(1)
where c is the transmitting speed of electromagnetic wave in underground, c0 (=3.0×108m/s)is the speed of light in vacuum, and ε is the relative permittivity of the soil.
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A processing unit provides B-Scope images by arranging the observed waveforms for the antenna positions on the ground surface. In the observed images, the horizontal axis represents the antenna unit position, and the vertical axis represents the propagation time of the electromagnetic wave, which is emitted in a wide angle with certain directional characteristics. When the antenna is scanned on the surface, the reflected wave from the same object can be received in the directivity range. If the pipe radius is regarded as a point, the reflected waveform constitutes a hyperbola shape. If not, the reflected waveform shows a pseudo-hyperbola that is different from the hyperbola. Therefore, the synthetic aperture method can be applied to the B-scope image, and constitutes a super resolution scheme.
Fig. 1 Scan lines
3 Problem Solution 3.1 Shortest Path Reflection Assumption In this work, the shortest path from the transmitter to the receiver via a point on object surface is geometrically calculated, and is assumed to be a substantial path. Therefore, for an object of flat plate, the path corresponds to specular reflections as shown in Fig.2.
Fig. 2 Specular reflection from plate plane 3.2 Problem Statement This method is based on the condition that the antenna unit is scanned in a specific direction on a ground surface more than one. In the following, taking an example of scanning in twice, the estimation procedure is described. However, it can be generalized for multiple scanning more than two times.
Variables to be used in the modeling are explained in Fig. 3. The scanning lines are shown by the axes Uk (k=0, 1, i.e., Uk means either U0 or U1), and they are assumed to be parallel to each other. The inclination angle of the buried pipe, φ, is defined by the rotation angle with respect to the Uk axis in the clockwise direction. The other rotation angle, i.e., the yaw angle, is defined by the rotation angle with respect to the Wk axis in the counter-clockwise direction.
At the k–th scanning, the horizontal displacement and the overburden are denoted by buk and bwk, respectively. Then, each of the coordinates of the observation points is indexed by the suffix i, and is denoted by uk,i (i =0,1,2,…). The propagating length at the observation point is denoted by lk,i. A reflecting point on the object is defined by a scaling factor with a unit vector a, and is denoted by sk,i
where the origin of the reflecting point is defined by the intersection between the Vk=0 plane and the buried pipe’s central line.
The problem is to estimate the unknown variables x = [θ, φ, r, bu0, bw0]
T from the propagating length data set lk,i (k=0, 1, and i=0,1,2,…) by applying a non-linear least squares method.
Fig. 3 Buried pipe and the coordinate systems
3.3 Propagating-Time Model Vectors bk and uk,i are given by
sin,coscos,sincos,, 321 aaaa (1)
cos,0,,, 321 rbbbbb wkukkkkk b (2)
0,0,1,, ,321, ikik uuuu u (3)
where
uuvuk tbtb 0tan (4)
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tancos0
vwwk
tbb (5)
If we denote the square of the length from the position uk,i to the buried pipe’s central line as fk,i, the the transmitting time, yk,i(x), is given by
rfc
lc
y
ik
ikik
,0
,0
,
2
2
x
(7)
where
2
,
2,
2,,
233,3
222,2
211,1
2,,,
cossin
coscoscossin
rbs
sbsu
basu
basubasu
sf
wkik
ikukikik
kik
kikkik
kikikik bau
(8) Thus, yk,i(x) can be modelled as a function of the unknown parameters, and Eq. (8) constitutes the regression equation for the non-linear least squares method.
Here, by using the constraint of ∂fk,i / ∂sk,i=0，that is,
0
222 33111,2
32
22
1,,
,
kkikikik
ik babaauaaass
f
(9)
the variable sk,i is given by
sincos
cossin,,
rbbus wkukikik
(10)
3.4 Non-Linear Least Squares Method The practical estimation procedure is described in the following. Let’s denote the observation equation as H, the covariance matrix of the observation vector as y. Based on the criterion that minimizes the squared sum of residuals of the observed values from the model-calculated values, the k-th correction vector is given by
kTkkTk
kk
Δ
k
Δ
ΔSk
yΣHHΣH
xxxΔ
yy
x
111
ˆminargˆ
where
k
n
k
k
n
kk
y
y
y
y
y
y
x
x
x
xyyΔy
ˆ
ˆ
ˆ
~
~
~ˆ~
1
1
0
1
1
0
)(ˆ1
1,1
1
1,11,11,11,1
1
1,1
1
1,11,11,11,1
1
0,1
1
0,10,10,10,1
0
1,0
0
1,01,01,01,0
0
1,0
0
1,01,01,01,0
0
0,0
0
0,00,00,00,0
jw
m
u
mmmm
wu
wu
w
n
u
nnnn
wu
wu
j
b
y
b
y
r
yyy
b
y
b
y
r
yyyb
y
b
y
r
yyyb
y
b
y
r
yyy
b
y
b
y
r
yyyb
y
b
y
r
yyy
xx
H
By using kxΔ ˆ , kx is recursively updated by
jjj xΔxx ˆˆˆ 1 (14)
while the covariance matrix of the errors of the kx is given by
11ˆ
HΣHΣ yxT
(15)
4 Experiment and Discussion 4.1 Experimental Method To confirm the effectiveness of the proposed algorithm, we carried out survey experiments for inspecting a stainless steel-made pipe being buried in soil medium. Fig. 4 shows the experimental setup: the antenna unit, the processing unit, and the rectangular tank (W: 1200mm * H: 600mm * D: 600mm) being fully filled with dried sand. The pipe was 50.8 mm in diameter r, 550 mm in length, and 1 mm in thickness. The relative permittivity of soil ε was calibrated as 3.24.
Then, we took data by setting the pipe direction as (θ, φ) = (0, 0), (10, 0), (0, 10), (10, 10). The values were approximate values, and the unit of them is deg.
Fig. 4 Experimental setup
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4.2 Experimental Results At each of the inclination angles, we scanned the antenna unit in the longitudinal direction of the tank. Fig. 5 shows some of the observed images. The peaks were extracted from the images, and are superimposed by the open circles in the images: the open circles are chosen with an interval of seven pixels. From these figures, it is, intuitively, seen to be very difficult to find the diameter and the direction of the pipe.
Furthermore, the observed values and the model-estimated ones are respectively shown by the open circles and by the full lines in Fig. 6. In the estimation, the observation errors of the propagation times are assumed to be identical and independent to one another for all the time-measurements, and, therefore, y=σy
2In where In is a n-th order of identity matrix. From the residual data of the measured times, we got σy
2=0.001ns2. Convergence is determined by a criterion that the ratio of the correction-vector norm to the updated-vector one becomes less than the predetermined threshold, i.e., 10-4 in the experiments: k
Δ and jΔ is
transformed into jw
j b 0ˆΔ and j
wj b 0
ˆΔ so as to make the magnitudes of angle and length match with each other.
(θ, φ)=(0, 0)
(θ, φ)=(10, 0)
(θ, φ)=(0, 10)
(θ, φ)=(10, 10)
Fig. 5 Peak-superimposed images
(θ, φ)=(0, 0)
(θ, φ)=(10, 0)
(θ, φ)=(0, 10)
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(θ, φ)=(10, 10)
Fig. 6 Experimental results of the theoretical values
fitted to the observed ones
Table 1 shows the results of estimations of the unknown parameters x = [θ, φ, r, bu0, bw0]
T. The lengths were estimated by the relative error of less than 10%, and the angles were also less than 10%. Thus, the unknown parameters ware successfully estimated.
For comparison, we also carried out another experiment by a conventional method: the pipe is assumed to be buried horizontally, and the scanning lines are to be perpendicular to the pipe [8]. As a result, it was confirmed that the errors by this work were much less than those by the conventional method.
Table 1 Estimated values by the proposed method
Angle θ φ
(θ , φ ) [deg] [deg] [mm] error [%] [mm] error [%] [mm] error [%]
(0, 0) 0.217 0.314 46.1 -9.3 443 2 267 4.7
(10, 0) 10.8 0.397 49.5 -2.6 473 0.6 268 3
(0, 10) -0.607 9.89 49.6 -2.4 435 -1 253 5.0
(10, 10) 11.0 9.97 49.0 -3.5 465 -0.5 240 4.6
r b u0 b w0
Table 2 Estimated values by the conventional
method
Angle
(θ , φ ) [mm] error [%] [mm] error [%] [mm] error [%](0, 0) 63.8 25.6 442 2 266 4.3(10, 0) 80.2 57.9 474 0.8 267 3
(0, 10) 60.1 18 435 -2 248 3
(10, 10) 61.1 20.3 475 2 234 2
r b u0 b w0
5 Conclusion In this paper, we examined a synthetic aperture method of the geometrical conditions of buried pipes. They were estimated so that the geometrical model-calculated propagation times are fit to the observed ones at all the antenna positions. Some experiments were carried out, and the effectiveness of the proposed algorithm was confirmed.
In the future, we are further directed to the development for shape estimation of the various types of buried objects. References: [1] T. Sakamoto and T. Sato, “A target shape
estimation algorithm for pulse radar systems based on boundary scattering transform,” IEICE Trans. Commun., vol. E87-B, no. 5, pp. 1357–1365, 2004.
[2] T. Sakamoto, “A fast algorithm for 3-dimensional imaging with UWB pulse radar systems,” IEICE Trans. Commun., vol. E90-B, no. 3,pp. 636–644, 2007.
[3] S. Kidera, T. Sakamoto, and T. Sato, “High-resolution and real-time UWB radar imaging algorithm with direct waveform compensations,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 11, pp. 3503–3513, Nov. 2008.
[4] S. Kidera, T. Sakamoto, and T. Sato, "Accurate UWB Radar Three-Dimensional Imaging Algorithm for a Complex Boundary Without Range Point Connections," IEEE Trans. Geosci. Remote Sence.vol.48, no.4, pp.1993~2004, April 2010.
[5] K. Takahashi and M. Sato, "Parametric Inversion Technique for Location of Cylindrical Structures by Cross-Hole Measurements," IEEE Trans. Geosci. Remote Sens., vol.44, no. 11, pp. 3348-3355, Nov. 2006.
[6] T. Kaneko, "Radar Image Processing for Locating Underground Linear Objects," IEICE Trans. Inf. & Syst., vol. E74-D, no. 10, pp. 3451-3458, Oct. 1991.
[7] Y. Nomura, Y. Naganuma, N. Kato, and Y. Sudo,“A Geometrical Analysis of Buried Flat-plates on Ground Penetrating Radar Images,” 2011 IEEE International Conference on Systems, Man, and Cybernetics, Oct. 9-12, 2011, No.10, pp.3317~3322, Anchorage, US
[8] H. Kotaki, Y. Nomura, H. Nishiguchi, and H. Matsui, "Buried Pipe Radius Estimation from Observed Image by GPR and Estimation Error Analysis," IEICE Trans. Commun., vol. J91-B, no. 9, pp. 1104-1112, Sep. 2008.
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ISBN: 978-1-61804-119-7 141