Directory UMM :Data Elmu:jurnal:J-a:Journal Of Applied Geophysics:Vol43.Issue2-4.2000:

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www.elsevier.nl_rlocate_rjappgeo

A matched-filter approach to wave migration

Carl Leuschen

a,)

_{, Richard Plumb}

b,1

Radar Systems and Remote Sensing Laboratory, Department of Electrical Engineering and Computer Science, The UniÕersity of Kansas, 2291 IrÕing Hill Road, Lawrence, KS, USA

Department of Electrical Engineering, State UniÕersity of New York at Binghamton, Binghamton, NY 13902-6000, USA

Received 20 October 1998; received in revised form 14 April 1999; accepted 13 May 1999

Abstract

Wave migration is a technique in which the reflectivity of the Earth is interpreted by extrapolating the fields measured on the surface into the ground. The motivation of this paper is to develop a generalized imaging algorithm based on a matched-filter that shows a mathematical connection between currently used migration techniques. The filter is determined by estimating the received signal when a specific test target exists in the ground. To keep the method general, a point scatterer is used as this target, while distributed objects are modeled without changing the filter characteristics by a collection of independent point scatterers. Also, the specific forms of the Green’s functions, which describe wave propagation in the ground, are not included in the formation of this approach leaving more freedom in the implementation. When the filter is applied to measured data of a monostatic survey, the resulting method becomes a forward scattering problem in which these data become time-reversed current sources. Next, specific forward scattering techniques are applied to this matched-filter approach and the resulting methods are compared to traditional migration techniques. In doing so, we find that the general form of most migration techniques can be shown using a matched-filter, while the major differences lie in the actual interpretation of the wave propagation that is used to implement the filter. The similarities of the matched-filter-based approaches to traditional techniques are used to show a connection and general overview of wave migration. Finally, these methods are applied to data collected over pipes buried in sand.q_{2000 Elsevier Science B.V. All} rights reserved.

Keywords: Ground-penetrating radar; Imaging; Migration; Scattering

1. Introduction

Wave migration, traditionally applied to

seis-mic applications and recently to

ground-penetrating radar, includes a number of

tech-)_{Corresponding author. Tel.: 785-864-7739; fax:}

1-785-843-7789; e-mail: [email protected]

Tel.: 1-607-777-4846; fax: 1-607-777-4464; e-mail: [email protected].

niques that are used to transform the scattered-field response from the x–t domain or image space into the x–z domain or object space

ŽClaerbout, 1971; Berkhout, 1981, 1982; Gazdag

and Sguazzero, 1984 . Due to the large

beamwidths of the receiving elements, images displayed in the x–t domain usually contain energy dispersed across much of the receiving array resulting in directional ambiguities. As a result, the only information that can be directly

Ž .

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interpreted from a single time response is range information when depth is desired. The transfor-mation into the x–z domain redistributes the energy in such a manner that these scattering events are migrated to their true location in space where depth can now be interpreted.

The main goal of this paper is to provide a

general overview of wave migration for

ground-penetrating radar using a matched-filter approach. The motivation is to provide a theo-retical connection to traditional seismic tech-niques so their use can be extended to radar applications. Using a generalized matched-filter

Ž .

definition Leuschen and Plumb, 1998 and spe-cific solutions and approximations to the wave equation, this paper outlines four methods for transforming a monostatic radar survey from the

x–t domain to the x–z domain. Also, three of

these methods are associated to traditional mi-gration algorithms, not only showing the con-nection between a matched-filter and migration but also a common link between existing tech-niques.

This paper will begin by examining the matched-filter technique and the general equa-tion for an image. The basic steps of the formu-lation will be presented along with the neces-sary approximations and assumptions. Next, four forward scattering techniques will be applied to this image equation. These methods include ba-sic spherical spreading, propagation in the f–k domain, a far-field approximation of the wave equation, and the finite-difference time-domain

ŽFDTD method. Finally, these methods will be.

related to existing techniques and applied to measurements collected over four air-filled PVC pipes buried in sand as a comparison.

2. The matched-filter technique

When a matched-filter is used to calculate the

Ž .

value at a pixel, the signal-to-noise ratio SNR is maximized when a specific target exists at

Ž .

that location Levanon, 1988 . In the formula-tion of the generalized matched-filter-based

im-age, this target is a point scatterer; while dis-tributed objects are approximated as a collection of independent point scatterers and multiple re-flections are ignored. The match-filter proce-dure involves correlating the received data,

Ž .

u r,t , with an estimated response from a point

Ž Y X

. Y

scatterer, h r ,r ,r,t , where r is the transmit-ter location, rX is the pixel location of the point scatterer, and r is the receiver location as shown by Fig. 1. The filter output is expressed as

S r

Ž .

X _s

HH

h r

Ž

Y,rX,r,t u r,

. Ž

_yt d rd t.

.

Ž .

1 This equation can be expressed as a convolu-tion.

X Y X _<

S r

Ž .

H

h r ,r ,r,t

Ž

.

_mu r,t d r

Ž

.

ts0

Ž .

2 We will begin by determining the

point-Ž Y X

scatter response, h r ,r ,r,t , in the frequency domain keeping in mind that the convolution

becomes multiplication. For an impulsive

source, the general form of the estimated

scat-Ž Y X

tered field due to a point target, H r ,r ,r,v , becomes

H r

Ž

Y,rX,r,v

.

X X X Y

sa G r,r ,

ˆ

r r

Ž

. Ž .

O r G r ,r ,i

Ž

.

a ,

ˆ

Ž .

3 where G and G are the incident and reflectedi r

dyadic Green’s functions for the electric field,

O is the object scattering matrix, and a is the

ˆ

polarization of the transmitting antenna, and a

ˆ

is the polarization of the receiving antenna. The object scattering matrix relates how the incident field induces current sources within the object, and if depolarization is ignored, the off-diagonal

Fig. 1. Configuration of wave scattering and collection.

Ž Y_. _Ž X_. _{Ž .}

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components become zero resulting in the iden-tity matrix. For a monostatic radar configura-tion,

rY_sr

Ž .

and

ˆ

i_sa .

ˆ

Ž .

Through reciprocity, Eq. 3 is simplified to

H r

Ž

X,r,v

.

_Ý

Ga2r

Ž

rX, r ,v

.

Ž .

where the subscript r corresponds to the direc-tion of the receiving antenna and a corresponds to three orthogonal spatial directions. Finally, through the use of the exploding reflector model

ŽLoewenthal et al., 1976 , the squaring opera-. Ž .

tion in Eq. 6 is expressed as

H r

Ž

X,r,v

.

Ý

Gar

Ž

2rX,2r,v

.

Ž .

Converting back to the time-domain, the general form of the image is expressed as

X X _<

S r

Ž .

_Ý

H

gar

Ž

2r ,2r,t

.

_mu r,

Ž

_yt d r

.

ts0.

Ž .

At this point, we disregard the mathematics and interpret the physical processes implied by this equation. The result is a forward scattering problem. First, the received signals become time-reversed sources. Next, these sources are placed at the receiver locations in the exact environment of the survey with all dimensions increased by a factor of two. The sources gener-ate waves and these waves are propaggener-ated until time equals zero. At this time, the spatial distri-bution of the scalar fields or the sum of the vector field components represents the image. This process can be accomplished using any forward scattering technique as will be shown later in this paper, but for now some properties

Ž X

of the filter, h r ,r,t , are addressed with their implications on the image.

Ž X _. _Ž X

First, h r ,r,t is not orthogonal with h r , r,1 2

. X X

t when r1 is not equal to r . This can be2

expressed mathematically as

h r

Ž

X,r,t h r

. Ž

X, r ,t d rd t

.

Ž .

HH

1 2

and implies that a non-zero output will occur at

rX2 when a scatterer exists at rX1. The result is range sidelobes, causing ambiguities of the ob-ject’s actual location. However, we can assume that this will only occur when rX2 is in close

X _< X X_<

proximity to r1 and as the distance r1_yr2

increases the output at rX₂ will become insignifi-cant. An example of this phenomenon is illus-trated by the finite bandwidth of the transmitted signal. Reflections in the x–t domain due to a scatterer at rX1 do not exist as a single event but are dispersed over time and tend to ring as they

Ž X

decay. Since h r ,r,t1 is not orthogonal to

Ž X

. X

h r ,r,t , values r₂ ₂ can be chosen such that a significant energy will occur at the output of

Ž .

Eq. 9 , and this ringing will be translated into the x–z domain.

Ž X

Second, the inner product of h r ,r,t is not normal or even constant for all values of rX. This is expressed as

h r

Ž

X,r,t h r

. Ž

X,r,t d rd t

.

_sf r

Ž .

X /constant

HH

Ž .

Ž X_.

and implies that an amplitude variation, f r , will exist over the image. Usually, this function is laterally independent and decreases with

Ž X

depth, and if f r can be approximated, its

effects can be eliminated by applying it as an inverse gain to the image.

3. Wave migration and the matched-filter

As stated earlier, migration, in a seismic or ground-penetrating radar sense, can be inter-preted as a redistribution of energy that was initially dispersed along a receiving array during the collection process to the actual or initial

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locations within the ground. In seismic tech-niques, migration is used to approximate the reflectivity at a point in space by estimating what the response would be given that both the transmitter and receiver were collocated at this point during the survey. Using this model, the reflectivity would be the received waveform at time equal to zero. Physically, this task is im-possible without digging into the ground, and, therefore, the fields that were collected at the surface are extrapolated back into the ground. These extrapolation processes that have been based on many ideas can usually be reduced to

Ž .

a forward scattering problem similar to Eq. 8 with the main differences lying in the physical interpretation of wave propagation. In the next sections, four different solutions of the wave equation are applied to the matched-filter image

Ž .

shown by Eq. 8 and resulting equations are compared to current migration techniques.

4. Diffraction summation

The simplest form of wave migration is diffraction summation or wavefront interference

migration Schneider, 1978; Gazdag and

Sguazzero, 1984 . This method uses a geometric interpretation of wave propagation rather than a solution of the wave equation. In this method, waves are assumed to propagate spherically with no dispersion, distortion, or attenuation. The function describing this process is expressed as

< X<

2 r_yr

g 2r,2r ,t

Ž

.

_sd

ž

t_y

/

Ž .

Substituting the delta function into Eq. 8 and integrating over time is equivalent to sampling the scattered-field response at times correspond-ing to the two-way travel distance divided by the velocity of the medium, Õ.

< X<

2 r_yr

S r

Ž .

H

u r,t

ž

/

d r

Ž .

The output in Eq. 12 is equivalent to the

conventional process of summing over a hyper-bola described by the two-way travel times from the receiver array to a pixel in space.

5. f–k migration

The fast f–k migration described by Stolt

Ž1978 is an efficient means of focusing the.

scattered-field response collected by a monos-tatic radar configuration back to the original location in the x–z domain via a two-dimen-sional Fourier transform. This method involves converting to the f–k domain, a translation ofx the energy into the k –k domain, and finally_x _z converting to x–z. In this section, migration from the f–kx to the k –kx z will be developed using a matched-filter definition based on elec-tromagnetic principles. In addition, the method will be expanded for the inclusion of a ground. The general form for the scalar two-dimen-sional matched-filter image in the frequency domain is expressed as

S x

Ž

X, zX;v

.

HH

U x ;

Ž

.

Ž

2 x

Ž

X_yx ,2 z

.

X;v

.

d xdv, 13

Ž .

where U x;v is the measured radar response

Ž .

at the location x on the surface z_s0 and

Ž . Ž .

S x, z;v is the image at location x, z . The

integral on the right-hand side is in the form of a convolution with respect to xX, and when transformed to the spatial frequency domain,

k , can be represented as multiplication.x

1 kXx

X X X U X

S k , z ;

Ž

x v

.

H

U k ;

Ž

x v

.

ž

,2 z ;v

/

2 2

Ž .

Assuming the medium is homogeneous in x, the Green’s function for the electric field is

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obtained analytically by solving the one-dimen-sional problem

2 2

Ž

Ž .

z _ykx

.

G k , z ;

Ž

x v

.

syjvmd

Ž .

z .

Ž .

15 For a medium that is also homogeneous in z, the conjugate of the Green’s function, shown in

Ž .

Eq. 14 , takes the form

kXx X U

ž

,2 z ;v

/

s _X ₂ 1r2

v k2

ž /

Õ 2

1r2

X 2 2

°

v k₂

¶

~

•

exp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

were Õ is the wave velocity in the medium.

Ž . Ž .

Substituting Eq. 16 into Eq. 14 and convert-ing to the spatial frequency domain, k , results_z in a delta function on the right-hand side.

S k

Ž

Xx, kXz;v

.

vm _X

H

_X ₂ 1r2U k ;

Ž

x v

.

v kx

ž /

Õ 2

1r2

X 2 2

v kx _X

d 2

ž /

_ykz dv

Ž .

Õ 2

0

This integration is performed by sampling v

when the argument of the delta function is zero and scaling by the inverse of the derivative of this argument with respect to v. By doing so, a

Fig. 2. One-dimensional interpretation of a source at the interface of a dielectric ground.

representation of the image in the spatial fre-quency domains, kXx and kXz, is expressed as

Õm Õ ₁_r₂

X X X X2 X2

S k , k

Ž

x z

.

_s U k ;

ž

x v_s kz _qkx

/

8 2

Ž .

The image shown by Eq. 18 is based on the one-dimensional free-space Green’s function,

Ž .

Eq. 16 , and is equivalent, excluding a scaling function, to the Stolt f–k migration. If the one-dimensional Green’s function is represented by only the exponential, the original form of the Stolt f–k migration can be reproduced exactly.

kXx X U

ž

,2 z ;v

/

1r2

X 2 2

°

v k2

¶

~

•

sexp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

S k

Ž

Xx, kXz

.

ÕkX

s _X₂ _X₂ 1r2

4 kx_qkz

Õ ₁_r₂

X X2 X2

U k ;

ž

x v_s kz _qkx

/

Ž .

20 2

The one-dimensional Green’s function above can also be modified to include the presence of a homogeneous ground. This function is ana-lyzed using two outward propagating waves that represent the fields in both the ground and the air. Each wave is characterized by a propagation constant and scaling factor shown in Fig. 2, where

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is the propagation constant in free space and

kzg_sv ´ ´ m

(

g 0 0

Ž .

22 is the propagation constant in the ground. The scaling factors, Aq _{and A}y_{, are determined by}

Ž .

evaluating Eq. 15 at the interface z_s0.

q y

A _sA _s

Ž .

k _qk

Ž

z zg

.

Ž .

Using this solution of Eq. 15 , the conjugate of

Ž .

the Green’s function, shown in Eq. 14 , takes the form

kXx X U

ž

, z ;v

/

s _X ₂ 1r2 _X ₂ 1r2

2 2

v kx v k2

y q y

ž /

1r2

X 2 2

°

v k2

¶

~

•

exp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

while the image is expressed as

S k

Ž

Xx, kXz

.

Õ2mkX

s ₂ ₂ 1r2

Õ Õ

X X2 X2 X2

4 k

½

z_q

ž /

kz _q

ž /

kx_ykx

5

c c

Õ ₁_r₂

X X2 X2

U k ;

ž

x v_s kz _qkx

/

Ž .

25 2

6. Far-field approximation and the Kirchhoff method

The general form for the matched-filter defi-nition of the image is expressed in the fre-quency domain as

X U T X

S r ,

Ž

.

HH

Ž

r ;v

.

a G r , r ,

ˆ

r r

Ž

.

X X Y

r G r , r ,v a d rdv. 26

Ž .

Ž

.

ˆ

Ž .

Fig. 3. Far-field approximation to monostatic scattering and collection. The configuration includes an x-directed

< X <

antenna, and a scatterer at a distance r _yr from the

antenna with only the u_x component of the scattering matrix.

In the monostatic radar configuration, both the transmitting and receiving antennas have the same location and like polarization.

r_srY

Ž .

ˆ

t_sa

ˆ

r_sx

ˆ

Ž .

28 Furthermore, using a far-field approximation, the induced currents within the scattering object can be limited only to the ux direction relative to the transmitting antenna as shown by Fig. 3. This is expressed mathematically in the object scattering matrix.

X X

O r

Ž .

_sOuu

Ž .

29 When the monostatic configuration and far-field

Ž .

approximation are applied to Eq. 26 , the im-age can be expressed with scalar values as

S r

_Ž

X;v

_.

H

_Ž

r;v

_.

G_u_x

_Ž

2rX,2r;v

_.

d r ,

_{Ž .}

30 where Gux is the ux-directed field due to an

x-directed source and all dimensions have been

increased by a factor of two due to the explod-ing reflector model.

An expression for G_u_x is obtained from the

free-space dyadic Green’s function Chew,

1990 ,

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where

4

G_u_x

_Ž

r,r

_.

_syjvm exp _yjkR sin

_{Ž .}

u_x , 4pR

Ž .

< X _<

R_sr _yr

Ž .

and ux is the angle between the x-axis and the vector rX_yr. An expression for the image is

Ž . Ž .

realized by substituting Eq. 32 into Eq. 30 . sin

Ž .

S r ;

Ž

.

HH

jvm

8pR

=exp

_yjk 2 R U

4 Ž

r,v

.

d rdv

Ž .

34 The integral over v can be divided into three operations on the measured response, U. The exponential represents sampling at _y2 R_rÕ

while the complex conjugate translates into a time reversal. The combination of these two operations is equivalent to sampling the response at 2 R_rÕ, which corresponds to the

two-way travel time. Finally, the jv term is interpreted as a derivative operation in the time-domain.

msin

Ž .

ux E 2 R

S r

Ž .

H

u r ;t

ž

/

d r .

Ž .

8pR Et Õ

Ž .

The form of Eq. 35 is similar to Kirchhoff migration,

cos

_{Ž .}

u_z E 2 R

S r

Ž .

H

u r ;t

ž

/

d r .

Ž .

2pRÕ Et Õ

Ž . Ž .

In Eq. 35 , the sin u term is a result of the

Fig. 4. Two-dimensional configuration of four 4-in. diame-ter air-filled PVC pipes buried in dry sand.

Fig. 5. Raw data with a linear gain applied.

radiation pattern of a infinitesimal dipole, while

Ž . Ž .

in Eq. 36 the cos u is a result of the solution to Kirchhoff integral.

7. FDTD reverse-time migration

The final type of propagation algorithm

de-Ž

scribed in this paper is the FDTD Taflove,

1995 . The FDTD method is an excellent tool for the simulation of wave propagation in the presence of complex environments, and,

there-Fig. 6. Resulting image from the diffraction summation approach.

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Fig. 7. Resulting image from f – k propagation.

fore, is used for wave migration in the presence of a ground and any other known objects. The method is a numerical technique in which Maxwell’s curl equations are solved in both space and time using a Taylor approximation.

E t

Ž

_qDt

.

_sE t

Ž .

_q ==H t

Ž .

´ Dt

H t

Ž

_qDt

.

_sH t

Ž .

_y ==E t

Ž .

A closed-form solution to Eq. 8 is not used. Rather, this equation is interpreted as a forward scattering problem and FDTD is used to

propa-Fig. 8. Resulting image from Stolt’s migration.

Fig. 9. Resulting image from the far-field approach.

gate the fields numerically. The basic method is outlined as reversing the data in time, increasing all dimensions by a factor of two, introducing

these time-reversed waveforms as current

sources into the FDTD lattice, and finally evalu-ating the fields at time equal to zero.

8. Experimental set up and results

An experiment consisting of four 10-cm di-ameter PVC pipes buried in dry sand was con-ducted in the sand-filled GPR test facility at The University of Kansas; it is 5=4.25=1.8 m

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Fig. 11. Resulting image from the FDTD-based reverse-time migration.

deep. All the pipes were oriented along the

y-axis simplifying the experiment into a

two-di-mensional space of x and z. The configuration shown by Fig. 4 indicates the relative locations of the PVC pipes, the concrete walls of the sandbox, and 30 monostatic antenna positions. The three pipes at the surface were purposely separated by both 20 and 30 cm center-to-center to investigate the lateral resolution of the algo-rithms, while the fourth pipe near the bottom is a permanent fixture of the box. The fourth pipe near the bottom is inline with the middle pipe near the surface and centered with respect to the synthetic radar aperture on the surface.

Data were recorded in a monostatic configu-ration. All measurements were collected using a Sensors and Software Pulse Ekko 1000

impul-sive radar system with a bandwidth and center frequency of about 900 MHz. The sampling interval of 20 ps was chosen to meet the stabil-ity criteria of the FDTD method, while 1500 samples were collected for each trace corre-sponding to a time interval of 30 ns. The traces shown in Fig. 5 were collected with y-polarized antennas at each of the 30 locations. A linear gain was applied to the data only to bring out some of the deeper events in the figure, and it should be noted that this gain was not applied to the data prior to the matched-filter processing.

Next, the data were migrated using each of the methods described above. Fig. 6 shows the results of applying a spherical-spreading-based matched-filter to the data. The implementation of both the f–k domain methods, matched-filter-based migration and the corresponding Stolt’s migration, are shown in Figs. 7 and 8, respectively. Kirchhoff migration is shown in Fig. 9, while the far-field matched-filter method is shown in Fig. 10. Finally, FDTD was used to implement the filter and the resulting image is shown in Fig. 11. Table 1 shows a brief sum-mary of all the techniques discussed in this paper including the domain transfers, equations governing wave propagation, and figure number of the resulting image.

9. Conclusions

As seen in the domain transfer function of Table 1, definite similarities exist between

Table 1

Overview of figure number, migration technique, transfer of domain, and equation governing wave propagation

Figure number and method Domain transform Wave propagation

Ž . Ž Ž< < ..

Fig. 6; spherical spreading matched-filter x–t™x–z d t_y r_yr _rÕ

diffraction summation

Ž . Ž . 4

Fig. 7; frequency–wave number matched-filter f–kx™k –kx z vm_r2 k expz _yjk rz

Fig. 8; Stolt’s migration f–kx™k –kx z exp_yjk rz

Ž . Ž Ž . .Ž . Ž Ž ..

Fig. 9; far-field approximation matched-filter x–t™x–z sinu _r8pR E_rEt d x;t_s2 R_rÕ

Žx-directed source.

Ž Ž . .Ž . Ž Ž ..

Fig. 10; Kirchhoff migration x–t™x–z cosu r2pRÕ ErEt d x;ts2 RrÕ

Ž . Ž . Ž . Ž . Ž .

Fig. 11; FDTD matched-filter x–t™x–z E t_qDt _sE t _q Dt_r´ ==H t

Ž . Ž . Ž . Ž .

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Stolt’s migration and the matched-filter ap-proach in the f–k domain, and also between Kirchhoff migration and the matched-filter ap-proach using the far-field approximation. Not only do these similarities show a strong connec-tion between the matched-filter approach and tradition techniques, but they also provide a better understanding of the processes involved with wave migration.

All the methods adequately resolved the buried pipes; however, the images produced using the time-domain methods contained pro-vided clearer images with less sidelobe

interfer-ence. Both the frequency-domain methods

showed interference occurring at the greater depths. This interference appears to be aliased wavefronts resulting from the transformation into the spatial frequency domain. Since the physical array exists over a finite length, the transformation into the spatial frequency do-main would have to exist over all wave num-bers. However, it is impossible to evaluate all spatial frequencies and the limits must be trun-cated. This truncation will cause unwanted sig-nals occurring in space and thus sidelobes. De-spite the poorer image quality, it should be noted that the frequency-domain approaches were the most computationally efficient. Since the wave propagation is accomplished simply by resampling the data in the frequency domain, most of the computations occurred in the Fourier transforms which were accomplished using a

Ž .

two-dimensional fast Fourier transform FFT . The time-domain methods required a summa-tion for each pixel which turned out to be more costly.

In this paper, some common migration tech-niques were investigated using a matched-filter

definition. The application of a matched-filter based on the scattering of a point scatterer was interpreted as a simple forward scattering prob-lem in which the received waveforms became time-reversed sources. This interpretation was realized with four different propagation equa-tions and then compared to traditional migration techniques. Finally, each method was applied to raw data of four pipes buried in dry sand.

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Prospecting 24 2 , 380–399.

Schneider, W.A., 1978. Integral formulation for migration in two and three dimensions. Geophysics 43, 49–76. Stolt, R.H., 1978. Migration by Fourier transform.

Geo-physics 43, 23–48.

Taflove, A., 1995. Computational Electrodynamics — The Finite-Difference Time-Domain Method. Artect House, Boston.

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obtained analytically by solving the one-dimen-sional problem

2 2

Ž

Ž .

z _ykx

.

G k , z ;

Ž

x v

.

syjvmd

Ž .

z .

Ž .

15 For a medium that is also homogeneous in z, the conjugate of the Green’s function, shown in

Ž .

Eq. 14 , takes the form

kXx X U

ž

,2 z ;v

/

s _X ₂ 1r2 2

v k2 2

ž /

Õ 2

1r2

X 2 2

°

v k₂

¶

~

•

exp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

were Õ is the wave velocity in the medium.

Ž . Ž .

Substituting Eq. 16 into Eq. 14 and convert-ing to the spatial frequency domain, k , results_z in a delta function on the right-hand side.

S k

Ž

Xx, kXz;v

.

vm _X

H

_X ₂ 1r2U k ;

Ž

x v

.

2 v kx

ž /

Õ 2

1r2

X 2 2

v kx _X

d 2

ž /

_ykz dv

Ž .

Õ 2

0

This integration is performed by sampling v when the argument of the delta function is zero and scaling by the inverse of the derivative of this argument with respect to v. By doing so, a

Fig. 2. One-dimensional interpretation of a source at the interface of a dielectric ground.

representation of the image in the spatial fre-quency domains, kXx and kXz, is expressed as

Õm Õ ₁_r₂

X X X X2 X2

S k , k

Ž

x z

.

_s U k ;

ž

x v_s kz _qkx

/

8 2

Ž .

The image shown by Eq. 18 is based on the one-dimensional free-space Green’s function,

Ž .

kXx X U

ž

,2 z ;v

/

1r2

X 2 2

°

v k2

¶

~

•

sexp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

S k

Ž

Xx, kXz

.

ÕkX

s _X₂ _X₂ 1r2 4 kx_qkz

Õ ₁_r₂

X X2 X2

U k ;

ž

x v_s kz _qkx

/

Ž .

(2)

is the propagation constant in free space and

kzg_sv ´ ´ m

(

g 0 0

Ž .

22 is the propagation constant in the ground. The scaling factors, Aq _{and A}y_{, are determined by}

Ž .

evaluating Eq. 15 at the interface z_s0. vm

q y

A _sA _s

Ž .

k _qk

Ž

z zg

.

Ž .

Using this solution of Eq. 15 , the conjugate of Ž . the Green’s function, shown in Eq. 14 , takes the form

kXx X U

ž

, z ;v

/

s _X ₂ 1r2 _X ₂ 1r2

2 2

v kx v k2

y q y

ž /

1r2

X 2 2

°

v k2

¶

~

•

exp j

_¢

ž /

2 z

_ß

Ž .

Õ 2

while the image is expressed as

S k

Ž

Xx, kXz

.

Õ2mkX

s ₂ ₂ 1r2

Õ Õ

X X2 X2 X2

4 k

½

z_q

ž /

kz _q

ž /

kx_ykx

5

c c

Õ ₁_r₂

X X2 X2

U k ;

ž

x v_s kz _qkx

/

Ž .

25 2

6. Far-field approximation and the Kirchhoff method

The general form for the matched-filter defi-nition of the image is expressed in the fre-quency domain as

X U T X

S r ,

Ž

.

HH

Ž

r ;v

.

a G r , r ,

ˆ

r r

Ž

.

X X Y

r G r , r ,v a d rdv. 26

Ž .

Ž

.

ˆ

Ž .

Fig. 3. Far-field approximation to monostatic scattering and collection. The configuration includes an x-directed

< X <

antenna, and a scatterer at a distance r _yr from the

antenna with only the u_x component of the scattering matrix.

In the monostatic radar configuration, both the transmitting and receiving antennas have the same location and like polarization.

r_srY

Ž .

ˆ

t_sa

ˆ

r_sx

ˆ

Ž .

Furthermore, using a far-field approximation, the induced currents within the scattering object can be limited only to the ux direction relative to the transmitting antenna as shown by Fig. 3. This is expressed mathematically in the object scattering matrix.

X X

O r

Ž .

_sOuu

Ž .

29 When the monostatic configuration and far-field

Ž .

approximation are applied to Eq. 26 , the im-age can be expressed with scalar values as

S r

_Ž

X;v

_.

H

_Ž

r;v

_.

G_u_x

_Ž

2rX,2r;v

_.

d r ,

_{Ž .}

30 where Gux is the ux-directed field due to an

x-directed source and all dimensions have been

increased by a factor of two due to the explod-ing reflector model.

An expression for G_u_x is obtained from the Ž free-space dyadic Green’s function Chew,

. 1990 ,

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where

4

G_u_x

_Ž

r,r

_.

_syjvm exp _yjkR sin

_{Ž .}

u_x , 4pR

Ž .

< X _<

R_sr _yr

Ž .

and ux is the angle between the x-axis and the vector rX_yr. An expression for the image is

Ž . Ž .

realized by substituting Eq. 32 into Eq. 30 . sin

Ž .

S r ;

Ž

.

HH

jvm

8pR

=exp

_yjk 2 R U

4 Ž

r,v

.

d rdv

Ž .

34 The integral over v can be divided into three operations on the measured response, U. The exponential represents sampling at _y2 R_rÕ

while the complex conjugate translates into a time reversal. The combination of these two operations is equivalent to sampling the response at 2 R_rÕ, which corresponds to the

two-way travel time. Finally, the jv term is interpreted as a derivative operation in the time-domain.

msin

Ž .

ux E 2 R

S r

Ž .

H

u r ;t

ž

/

d r .

Ž .

8pR Et Õ

Ž .

The form of Eq. 35 is similar to Kirchhoff migration,

cos

_{Ž .}

u_z E 2 R

S r

Ž .

H

u r ;t

ž

/

d r .

Ž .

2pRÕ Et Õ

Ž . Ž .

In Eq. 35 , the sin u term is a result of the

Fig. 4. Two-dimensional configuration of four 4-in. diame-ter air-filled PVC pipes buried in dry sand.

Fig. 5. Raw data with a linear gain applied.

radiation pattern of a infinitesimal dipole, while

Ž . Ž .

in Eq. 36 the cos u is a result of the solution to Kirchhoff integral.

7. FDTD reverse-time migration

The final type of propagation algorithm de-Ž

scribed in this paper is the FDTD Taflove, .

1995 . The FDTD method is an excellent tool for the simulation of wave propagation in the presence of complex environments, and,

there-Fig. 6. Resulting image from the diffraction summation approach.

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Fig. 7. Resulting image from f – k propagation.

E t

Ž

_qDt

.

_sE t

Ž .

_q ==H t

Ž .

37 ´

H t

Ž

_qDt

.

_sH t

Ž .

_y ==E t

Ž .

38 m

Ž .

A closed-form solution to Eq. 8 is not used. Rather, this equation is interpreted as a forward scattering problem and FDTD is used to

propa-Fig. 8. Resulting image from Stolt’s migration.

Fig. 9. Resulting image from the far-field approach.

gate the fields numerically. The basic method is outlined as reversing the data in time, increasing all dimensions by a factor of two, introducing these time-reversed waveforms as current sources into the FDTD lattice, and finally evalu-ating the fields at time equal to zero.

8. Experimental set up and results

An experiment consisting of four 10-cm di-ameter PVC pipes buried in dry sand was con-ducted in the sand-filled GPR test facility at The University of Kansas; it is 5=4.25=1.8 m

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Fig. 11. Resulting image from the FDTD-based reverse-time migration.

deep. All the pipes were oriented along the

y-axis simplifying the experiment into a

Data were recorded in a monostatic configu-ration. All measurements were collected using a Sensors and Software Pulse Ekko 1000

9. Conclusions

As seen in the domain transfer function of Table 1, definite similarities exist between

Table 1

Overview of figure number, migration technique, transfer of domain, and equation governing wave propagation Figure number and method Domain transform Wave propagation

Ž . Ž Ž< < ..

Fig. 6; spherical spreading matched-filter x–t™x–z d t_y r_yr _rÕ

diffraction summation

Ž . Ž . 4

Fig. 7; frequency–wave number matched-filter f–kx™k –kx z vm_r2 k expz _yjk rz

Fig. 8; Stolt’s migration f–kx™k –kx z exp_yjk rz

Ž . Ž Ž . .Ž . Ž Ž ..

Fig. 9; far-field approximation matched-filter x–t™x–z sinu _r8pR E_rEt d x;t_s2 R_rÕ

Žx-directed source.

Ž Ž . .Ž . Ž Ž ..

Fig. 10; Kirchhoff migration x–t™x–z cosu r2pRÕ ErEt d x;ts2 RrÕ

Ž . Ž . Ž . Ž . Ž .

Fig. 11; FDTD matched-filter x–t™x–z E t_qDt _sE t _q Dt_r´ ==H t

Ž . Ž . Ž . Ž .

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All the methods adequately resolved the buried pipes; however, the images produced using the time-domain methods contained pro-vided clearer images with less sidelobe interfer-ence. Both the frequency-domain methods showed interference occurring at the greater depths. This interference appears to be aliased wavefronts resulting from the transformation into the spatial frequency domain. Since the physical array exists over a finite length, the transformation into the spatial frequency do-main would have to exist over all wave num-bers. However, it is impossible to evaluate all spatial frequencies and the limits must be trun-cated. This truncation will cause unwanted sig-nals occurring in space and thus sidelobes. De-spite the poorer image quality, it should be noted that the frequency-domain approaches were the most computationally efficient. Since the wave propagation is accomplished simply by resampling the data in the frequency domain, most of the computations occurred in the Fourier transforms which were accomplished using a

Ž .

two-dimensional fast Fourier transform FFT . The time-domain methods required a summa-tion for each pixel which turned out to be more costly.

In this paper, some common migration tech-niques were investigated using a matched-filter

References

Berkhout, A.J., 1981. Wave field extrapolation techniques in seismic migration, a tutorial. Geophysics 46, 1638– 1656.

Berkhout, A.J., 1982. Seismic Migration — Imaging of Acoustic Energy by Wave Field Extrapolation: A. The-oretical Aspects. Elsevier, Amsterdam.

Chew, W.C., 1990. Waves and Fields in Inhomogeneous Media. Van Nostrand-Reinhold, New York.

Claerbout, J.F., 1971. Toward a unified theory of reflector mapping. Geophysics 36, 467–481.

Gazdag, J., Sguazzero, P., 1984. Migration of seismic data.

Ž .

Proceedings of the IEEE 72 10 , 1302–1315. Leuschen, C., Plumb, R., 1998. A matched-filter-based

reverse-time-migrating algorithm for ground-penetrat-ing Radar Data. Review for IEEE Transactions on Geoscience and Remote Sensing.

N. Levanon, 1998. Radar Principles. Wiley, New York, 1988.

Loewenthal, D., Lu, L., Roberson, R., Sherwood, J., 1976. The wave equation applied to migration. Geophysical

Ž .

Prospecting 24 2 , 380–399.

Schneider, W.A., 1978. Integral formulation for migration in two and three dimensions. Geophysics 43, 49–76. Stolt, R.H., 1978. Migration by Fourier transform.

Geo-physics 43, 23–48.

Taflove, A., 1995. Computational Electrodynamics — The Finite-Difference Time-Domain Method. Artect House, Boston.