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GEOPHYSICS, VOL. 65, NO. 1 (JANUARY-FEBRUARY 2000); P. 113125, 11 FIGS., 1 TABLE.

Synthesis of amplitude-versus-offset variationsin ground-penetrating radar data

Xiaoxian Zeng, George A. McMechan, and Tong Xu

ABSTRACT

To evaluatethe importance of amplitude-versus-offsetinformation in the interpretation of ground-penetratingradar (GPR) data, GPR reflections are synthesized asa function of antenna separation using a 2.5-D finite-difference solution of Maxwells equations. The conduc-tivity, the complex dielectric permittivity, and the com-plex magnetic permeability are varied systematically innine suites of horizontally layered models. The sourceused is a horizontal transverse-electric dipole situatedat the air-earth interface. Cole-Cole relaxation mecha-nisms define the frequency dependence of the media.Reflection magnitudes and their variations with antennaseparation differ substantially, depending on the con-trast in electromagnetic properties that caused the re-

flection. The spectral character of the dielectricand mag-netic relaxations produces only second-order variationsin reflection coefficients compared with those associatedwith contrasts in permittivity, conductivity, and perme-ability, so they may not be separable even when theyare detected. In typical earth materials, attenuation ofpropagating GPR waves is influenced most strongly byconductivity, followed by dielectric relaxation, followedby magnetic relaxation. A pervasive feature of the sim-ulated responses is a locally high amplitude associatedwiththe critical incidentangle at the air-earth interfaceinthe antenna radiationpattern. Fullwavefieldsimulationsof two field data sets from a fluvial/eolian environment

are able to reproduce the main amplitude behaviors ob-served in the data.

INTRODUCTION

Amplitude information rarely is used explicitly in interpre-tation of ground-penetrating radar (GPR) data because the

Manuscript received by the Editor September 2, 1997; Revised manuscript received May 28, 1999.The University of Texas at Dallas, Center for Lithospheric Studies, P.O. Box 830688 (FA31), Richardson, Texas 75083-0688. E-mail: [email protected]; [email protected] The University of Texas at Dallas, Center for Lithospheric Studies, P.O. Box 830688 (FA31), Richardson, Texas 75083-0688; currentlyTexaco Exploration and Production Technology, 3901 Briarpark, Houston, Texas 77042. E- mail: [email protected] 2000 Society of Exploration Geophysicists. All rights reserved.

numerical modeling tools required for systematic study havebecome available only recently. Including analysis of GPR am-plitude data in environmental and engineering site evaluations

and in geologic applications potentially constrains the rangeof possible interpretations. One way to investigate the rela-tions between the recorded amplitudes and the distributionof electromagnetic properties that produced them is throughnumerical simulations for simple models.

Numerical modeling algorithms for two- or three-dimen-sional (2-D or 3-D) electromagnetic responses at GPR fre-quencies have been presented by a number of authors. Thesealgorithms range from ray-based methods (Goodman, 1994;Cai and McMechan, 1995; Powers and Olhoeft, 1996) to inte-gral equation solutions (Xiong and Tripp, 1997a, b) to Fouriermethods (Powers and Olhoeft, 1994, 1995; Zeng et al., 1995;Carcione, 1996) and time-domain finite differencing (Yee,

1966; Luebbers et al., 1990; Maloney et al., 1990; Tirkas andBalanis, 1992; Livelybrooks and Fullagar, 1994; Roberts andDaniels, 1994; Casper and Kung, 1996; Wang and Tripp, 1996;Xu and McMechan, 1997; Bergmann et al., 1998). In most ofthese papers, attenuation of GPR waves is considered to beproduced only by electrical conductivity, because it is usuallythe primary attenuation mechanism (ignoring dielectric andmagnetic relaxations), or the relaxations are included in an ef-fective conductivity that is proportional to frequency. For ex-amples of the latter, see Turner and Siggins (1994) and Casperand Kung (1996).

Previous studies have considered numerical and analytic so-lutions for normal incidencereflection coefficients as a functionof frequency (Luebbers et al., 1990; Luebbers, 1992), scatter-

ing by targets of specific geometry (Britt, 1989; Carcione, 1996;Xu and McMechan, 1997), or resonant modes in thin layers(Rossiter et al., 1975). A few examples of multioffset GPRsimulations exist (e.g., Xiong and Tripp, 1997a, b). However,we could find no systematic study of reflections from an in-terface as a function of offset, except for variations of singleparameters such as conductivity (e.g., Turner, 1992). As the

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114 Zeng et al.

requisite numerical tools are now available for more compre-hensive studies of GPR amplitude, the time is right to performsuch calculations. The results of the numerical simulations pre-sented below show that the use of the offset dependence ofamplitude is potentially a useful diagnostic element in inter-pretation of GPR data.

For the purpose of understanding amplitude-versus-offset

(AVO) data, it is useful to consider separately the effects ofthe complex dielectric permittivity (), the complex magneticpermeability (), and the conductivity () on the propaga-tion and reflection of GPR waves. We represent the frequencydependence of and by using the four-parameter empir-ical description of Cole and Cole (1941). The Cole-Cole pa-rameterization is common in laboratory studies of frequency-dependent electric properties (e.g., Olhoeft and Capron, 1993)but has been used only rarely in numerical modeling of GPRdata; exceptions are the algorithms of Powers and Olhoeft(1996), Powers (1996), and Xu and McMechan (1997). TheCole-Cole parameterization potentially allows AVO analysisof common-reflection-point GPR gathers for interpretation ofelectromagnetic material properties, following proceduressim-

ilar to those used forextraction of elastic or viscoelastic proper-ties in reflectionseismology (Castagna and Backus, 1993; Allenand Peddy, 1993). Our goal in this paper is to investigate andanalyze the main AVO behaviors in GPR data through a lim-ited suite of synthetic examples, and thereby to determine therelative effects of the various electromagnetic properties onobserved amplitudes.

NUMERICAL MODELING

Model parameterization

For numerical modeling, we use the 2.5-D staggered-gridfinite-difference solution of Maxwells equations described by

Xu and McMechan (1997). This scheme explicitly includes at-tenuation and dispersion associated with dielectric and mag-netic relaxations as well as conductivity. Relaxations are mod-eled usingonlytwo Cole-Colefunctionsfor each material inthemodel; this corresponds to only one distribution for dielectricrelaxation and one for magnetic relaxation in each material.

The complex dielectric permittivity is expressed through theempirical Cole and Cole (1941) formula

() = () + () = 0 +0(l )

1+ (i/)(1)

where and are the real and imaginary parts of; 0 is thedielectric permittivity in a vacuum (0

=8.854

1012 F/m);

0 1; i =1; is the relaxation radial frequency; andl and are the relative dielectric permittivities at very lowand very high frequencies, respectively. defines the widthof the frequency-dependent transition between the high- andlow-frequency limits of () (Figure 1). = 1 correspondsto a single Debye relaxation mechanism; < 1 correspondsto a distribution of mechanisms centered around . The fourparameters l , , , and describe the dependence of onthe radial frequency of the source . The real and imaginaryparts of the relative dielectric permittivity (/0) for a typicalearth material are shown in Figure 1.

In some soils, such as iron-enriched sand, the magnetic per-meability is also a complex, frequency-dependent parameter.

The behavior of the magnetic relaxation can be treated usingthe same empirical form as for dielectric relaxation (Olhoeft,1972; Olhoeft and Strangway, 1974; Olhoeft and Capron, 1994)with the magnetic permeability variables , , , 0, l , ,, and replacing the dielectric permittivity variables

, ,, 0, l , , , and , respectively, in equation (1), where0 = 4 107 H/m. The relaxation frequencies used for all

the models are (radians/s)= 2 (radians)125 106

(Hz),and = 2 67 106 (radians/s), which are representativeof values measured in soils (Olhoeft and Capron, 1993). Inthe finite-difference solution, the relaxation mechanisms areincluded through memory variables (Carcione, 1996; Xu andMcMechan, 1997; Bergmann et al., 1998) which replace convo-lution between the electromagnetic fields and the time-varyingmaterial properties. In a material with no relaxation mecha-nisms, the effective relative permittivity and permeability areconstant, with values (l )/2and(l )/2, respectively,as obtained by setting = 0 in equation (1). Attenuation stillmay occur if the medium is conductive. Most of the modelsconsidered below have at least one element with = 0.

FIG. 1. Frequencydependence of the(a) real and(b) imaginaryparts of the complex relative dielectric permittivity (/0) ofa typical earth material. The curves are given by the Cole-Coleequation (1) using l = 9.0, = 6.0, and = 2 radians 125 MHz. The label on each curve pair is the correspondingvalue of. The real part of (

/0) defines the velocity disper-sion; the ratio of the imaginary part to the real part is the losstangent, which defines the attenuation. The maximum attenu-ation occurs near (Gueguen and Palciauskas, 1994).


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Dipole source definition

We consider only horizontal electric dipole antennae placedat the air-earth interface (the free surface) for both trans-mitters and receivers. These dipoles are oriented transverseto the (x z) plane of the computational grid (with long axesperpendicular to the profile, in the y direction). The dipolesource is defined by specifying a nonzero component of source

current (Jy) at the source location (Xu and McMechan, 1997);the dipole antenna radiation pattern is a consequence of thefree-surface boundary conditions (e.g., Annan, 1973; Enghetaet al., 1982) and thus implicitly changes as the (complex)properties of the uppermost earth layer change (Turner, 1994).The maximum radiation is in the (x z) plane, at the criticalangle ofc= sin1(n1), where n is the refractive index of theearth material below the air, and angle c is measured fromvertical in the (x z) plane. At angles beyond c, a criticallyrefracted head wave travels along the free surface with the airvelocity (Clough, 1976).

Superimposing 2-D solutions for different horizontal wave-numbers (ky) simulates 3-D geometric spreading for a finite-

length dipole oriented perpendicular to the (x z) plane (Xuand McMechan, 1997). We used 0 ky 4, with ky = 1, assuggested by Livelybrooks and Fullagar (1994). With the pa-rameters listed below, a common-source gather (for one ky) isproduced in about fiveminutes in a single 167-MHz Ultrasparcprocessor.

Simulation parameters

The model grid used was 230 450 points with a 2-cm in-crement (4.6 m in depth and 9.0 m in the horizontal direction),excluding1 point on allfouredges usedfor implementingMurs(1981) second-order absorbing boundary condition. The mod-els were run for 1800 time steps with an increment of 0.04 ns,

for a total time of 72 ns. These space and time steps satisfy thestability and grid-dispersion criteria (Petropoulos, 1994) in allthe models used. At least 10 grid points per wavelength arerecommended (e.g., Kunz and Luebbers, 1993); our examplesconsistently exceed this at the dominant frequency, usually bymore than a factor of two. Except where stated otherwise, thesimulations below are for a band-limited (Ricker) source timewavelet with a dominant frequency of 200 MHz; where higherfrequencies are used, the computation parameters are scaledproportionally.

Processing and plotting format

To remove any problems caused by direct waves interferingwith the target reflection, the response of a homogeneous half-space with the properties of the upper layer is subtracted fromthe simulation to isolate the reflected phases.

For display, each reflection is flattened (approximately) intime by a simple shift of each trace. The shift is defined bythe predicted normal moveout curve for a reflection createdat 1.2-m depth using a velocity of 0.1074 m/ns (correspondingto a relative permittivity of 7.5, a relative permeability of 1.0,and no dispersion). This is done only for display purposes, butbecause it is identical for all examples, the relative moveout ispreserved. A normal moveout correction (as typically done inseismic data processing) is not done because this would stretchthe wavelets and hence distort the amplitude measurements.

For each trace, the AVO measurement consists of summingthe square of the electric field amplitude of the reflection andits associated arrivals (e.g., the small head wave produced atthe free surface). Then the square root of the sum is taken toget back to amplitude, which is plotted as a function of offset(Demirbaget al., 1993). This process results in an exceptionallyrobust measurement, but the polarity information is lost. Thus,

polarity and waveshape (phase) information is described in thetext where appropriate.

SYNTHESIS AND ANALYSIS OF GPR AVO CURVES

In this section, we use a flat-layered earth model to examinethe basic AVOresponses in the absence of amplitude variationsresultingfrom model geometry. Then thefollowingsectioncon-tains field-data examples in which the primary behaviors arestill visible but are distorted by various lateral variations inproperties and by wavefield focusing and defocusing. All com-puted responses and the extracted AVO curves contain all theeffects of the properties of the model and of the antenna radi-ation pattern.

Nine suites of models are used to investigate amplitude-versus-offset behavior; all models have the same flat-layer ge-ometry consisting of an air layer above two layers of variousearth materials. Each suite of models (and AVO responses) isobtained by perturbing a single parameter in either the layeror the half-space, while keeping all other parameters fixed. Indescribing these models, all permittivity and permeability val-ues are relative (i.e., = /0 and =/0). The propertiesof the air layer (= 1.0; = 1.0; = 0.0) are the same in allmodels and so will not be considered further. The two earthlayers are an upper layer of 1.2-m thickness and an underlyinghalf-space; these are designated layers 1 and 2, respectively,and their properties are given in Table 1 for all models.

Traces are computed for an aperture along the free surfaceof 0 to 5 m that is centered between the edges of the compu-tational grid. This geometry corresponds to incidence anglesat the reflector of 0.0 to 64.34. Because the reflector is flat,the AVO behavior in a common-source gather is equivalent tothat in a common-reflection-point gather, but it is much lessexpensive to compute.

This section is divided into three main parts. First, we varythe properties of the half-space beneath the reflector to pro-duce suites of AVO curves to illustrate how these propertiesaffect reflectivity. Then we vary the properties of the layer be-tween the reflector and the free surface to illustrate the ef-fects of the properties of that layer on propagation through it.Finally, we briefly consider responses as a function of sourcefrequency. Some of the scenarios considered go beyond therange of typical geologic material properties but are includedfor completeness because they address the important situa-tion of buried anomalies that are the targets of engineeringand environmental site evaluations. For example, a mediumwith a high normally would have high also, but varying asingle parameter is more instructive regarding how each pa-rameter affects the AVO response. Reflection magnitudes andAVO curve shapes differ markedly, depending on the contrastin electromagnetic properties that caused the reflection andonthe properties of the overlying material. At any given offset,the geometric spreading is identical for all models and so doesnot contribute to AVO differences. Differences between the


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116 Zeng et al.

observed and reference AVO responses commonly are usedin seismic AVO studies (e.g. Chiburis, 1987; Allen and Peddy,1993).

Effects of the lower-layer properties on AVO observations

Figure 2 shows four common-source GPR gathers computed

for a range of dielectric permittivity values in the lower layer(model1 in Table 1).Reflection amplitudes increase as thecon-trast in permittivity increases, and the phase of the reflectedarrival is reversed when the sign of the contrast is reversed(e.g., Figures 3a and 3d). The reflections in Figures 3c and 3dcontain patterns of amplitude and phase changes that are as-sociated with critical reflections and head waves, even thoughthe reflectorshave negative impedance contrasts. These criticalreflections are generated at the free surface, not at the subsur-face reflector, so their interpretation will be in error if theyare attributed to a subsurface reflector. An additional postcrit-ical reflection and head wave are produced at the subsurfacereflector when the impedance contrast is positive (Figure 3a).

The AVO plot in Figure 4a shows that the position of thehigh-amplitude peak associated with critical reflection at thefree surface is independent of the properties of the bottomlayer and remains stationary at about 1.9-m offset. However,the critical reflection produced by a low-to-high impedancecontrast at the reflector moves to progressively shorter offsetwith increasing impedance of the lower layer. Such observa-tions of critical angles at internal reflectorsare rare, but notim-possible, in field GPR data as permittivity generally increaseswith depth. The amplitude peaks in the other plots in Figure 4also are associated with the free surface.

Figures 4b and 4c show the AVO behavior as a functionof conductivity and permeability in the lower layer (models 2and 3, respectively, in Table 1). Normal-incidence reflections

from both conductivity and magnetic permeability contrastswere illustrated previously by Lazaro-Mancilla and Gomez-Trevi no (1994), who concluded that such reflections could besimilar in magnitude to those generated by contrasts in dielec-tric permittivity. Figures 4b and 4c extend the validity of thisconclusion to nonnormal incidence although the relative am-plitude decrease with increasing offset is more rapid for per-meability than for conductivity. Although the common-sourcegathers are not shown for these examples, the polarity, waveletshape, and overall behavior of the conductive model responsesare similar to those of the large-permittivity models (e.g., Fig-

Table 1. Electromagnetic properties of the models used for the examples. The models also include an air layer above layer 1. For, only the average value is given; when isnot0, l and are20%, respectively, of the average values. For, only the averagevalue is given; when is not 0, l and are13%, respectively, of the average values.

Layer 1 Layer 2

Model

1 7.5 0. 1.0 0. 0. 2.056.0 0. 1.0 0. 0.2 7.5 0. 1.0 0. 0. 7.5 0. 1.0 0. 0.1.03 7.5 0. 1.0 0. 0. 7.5 0. 1.02.8 0. 0.4 7.5 0. 1.0 0. 0. 7.5 0.1.0 1.0 0. 0.5 7.5 0. 1.15 0. 0. 7.5 0. 1.15 0.1.0 0.6 7.5 0. 1.0 0. 00.05 20.5 0. 1.0 0. 0.7 7.5 0. 1.02.8 0. 0. 20.5 0. 1.0 0. 0.8 7.5 0.1.0 1.0 0. 0. 20.5 0. 1.0 0. 0.9 7.5 0. 1.15 0.1.0 0. 20.5 0. 1.15 0. 0.

ures 3c and3d),whilethe permeability modelresponses resem-ble the low-permittivity model responses (e.g., Figures 3a and3b). This is expected because of the reciprocal positions of (or ) and in the expression for intrinsic electric impedance(e.g., Belanis, 1989):

Z= i

+i

. (2)

The polarity of reflections from contrasts in any of, , or

alone cannot be predicted without knowledge of the values ofall the other parameters in Z, including .

The reflections from contrasts in dielectric and magnetic re-laxations alone are not shown, but they have characteristicssimilar to reflections from small positive contrasts in permit-tivity and permeability, respectively. The permittivity and per-meability in the impedance (equation 2) are complex (equa-tion 1), and so the effective impedance depends on both theimaginary and real parts of and . If = 0 and is purereal, the magnitude of the impedance corresponding to a Cole-Cole complex dielectric permittivity is (von Hipple, 1995)

|Z| =

1+ tan2 (3)

where tan is the loss tangent defined by the ratio /. For

= 0 in equation (1), = 0,sothereisnoloss,and |Z| takes themaximum value. As tan , increases, |Z| decreases. Similarly,if = 0 and is pure real, the magnitude of the impedancecorresponding to a Cole-Cole complex magnetic permeabilityis (von Hipple, 1995)

|Z| =

1+ tan2 (4)

where tan is the loss tangent defined by the ratio /. As

tan increases, |Z| increases.Figures 4d and4e contain theAVOcurves formodels 4 and5,

respectively (Table 1). In both models, increasing (equa-tion1) producesincreasing contrastin impedance derived from (or ) and hence increasing reflection amplitude, but thereflectionmagnitudes arevery small, evenfor = 1.Asaconse-quence of the reciprocal behavior described above, the sign ofthe reflection coefficient obtained by increasing tan through is opposite to that obtained by increasing tan through .It is unlikely that these small-amplitude contributions to re-flectivity can be distinguished in the AVO analysis of field data


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(especially in the presence of contrasts in , , and/or in .This is not the case, however, for the attenuation of propagat-ing waves associated with dielectric and magnetic relaxations(see below).

Effects of upper- (overburden) layer properties on AVO

observations

In this section, we consider the AVO effects caused bychanges in the electromagnetic properties of the layer between

FIG.2. Common-source GPRgathersfor models withcontrastsin dielectric permittivity (model 1 of Table 1). The dielectricpermittivityof theupper layer is fixed at 7.5; forthe lower layer,it is (a) 2.0,(b) 6.0, (c) 20.5, and (d) 56.0. In(a),R is the primaryreflection from the reflector at 1.2-m depth, A is the direct airwave, G is thedirect ground wave,and M is thefirstfree-surfacemultiple. Thedirect-wave amplitudes may be inaccurate withina few grid points of the source (Roberts, 1994). The (constant)amplitude scaling is the same for all panels. Large amplitudesare clipped for clarity.

the reflector and the free surface. Changes in these propertiesproduce complex interactions among three factors: the cou-pling and radiation pattern of the antennae at the free surface;the attenuation, dispersion, and propagation time through thelayer; and the reflectioncoefficient at the reflector. We havenotattempted to simulate the effects associated with local varia-tions in antenna orientationor height, or with surficial geologic

conditions typically present in field data.Because we are most interested in the relative effects ofthe three attenuation mechanisms on the waves propagatingthrough theupper layer, allthe models(6, 7, 8, and 9 in Table 1)maintain the same contrast in l and across the reflector toensure that all models produce visible reflections. The differ-ences between the resulting AVO curves are caused by thedifferences in the overburden attenuations. The effective rela-tive dielectric permittivity in the lower layer is 20.5. This valuelies near the center of the range of values used for the lowerlayer in the previous section, and so it provides a tie betweentheresults presentedabove and those in this section. The effectof permittivity contrasts already have been considered aboveand so are not repeated here.

Figure 5 shows the flattened reflections for three values ofin the upper layer, while keeping all parameters of the lowerlayer fixed (model 6 of Table 1). The plots in Figure 5b arefor = 0.01 S/m, which often is considered the upper limit forobtaining useful GPR data. Similarly, Figures 6a, 6b, and 6ccontain flattened reflections for three values of (relative) in

FIG. 3. Preprocessed and flattened reflections extracted fromthecommon-source gathers in Figure 2 (for model1 in Table 1).Preprocessing includes subtraction of the direct air and groundwaves. The dielectric permittivity of the upper layer is fixed at7.5; for the lower layer it is (a) 2.0, (b) 6.0, (c) 20.5, and (d) 56.0.Amplitude scaling is the same for all panels. Large amplitudesare clipped for clarity.


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the upper layer, while keeping fixed at 1.0 in the lower layer(model 7 of Table 1).

Amplitude attenuation is very sensitive to ; note that theamplitude scale on the AVO plot of Figure 7a is a log scale. At= 0.01 S/m, theamplitude is reduced from thenonconductivecase by a factor of about 10 at zero offset, and by nearly 100 at5-m offset (Figure 7a).

Perhaps the most interesting effect of increasing is thatthe propagation velocity v decreases (Figure 6). If = 0 and

FIG. 4. AVOcurves for reflections from contrasts in (a) relative dielectricpermittivity , (b) electrical conductivity, (c) relative magnetic permeability , (d), dielectric relaxation exponent , and (e) magnetic relaxationexponent . These correspond to model sets 1, 2, 3, 4, and 5 of Table 1, respectively. The label on each curvegives the value of the property perturbed in the lower layer; quantities in (a), (c), (d), and (e) are dimensionless,and in (b), is in S/m. In (a), the light, solid lines are for a negative impedance contrast (high over low) andthe heavy dashed lines are for a positive contrast (low over high); the corresponding reflections have opposite

polarity.

both and are small, then the propagation velocity is givenby (e.g. von Hipple, 1954; Gueguen and Palciauskas, 1994)

v = 1

2cos

1+ cos . (5)

The loss tangent tan (= /+/) empirically describesthe net effect of the electromagnetic properties that producevelocity dispersion as a function of frequency (as described


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below). If increases sufficiently in the upper layer, a low-velocity waveguide is produced [see equation (5)]. This waveg-uide has a critical angle of propagation (and hence also a headwave) at the lower boundary as well as at the free surface(B and F, respectively, in Figure 6d). Head wave F propagateswith the velocity of air; head wave B propagates with the ve-locity of the lower half-space. The AVO behavior (see curve A

in Figure 7b) is correspondingly complicated. The position ofthe peak amplitude of the AVO curves (Figure 7b) also shiftssystematically with as the resulting velocity changes [equa-tion (5)] shift the critical angle at the free surface.

The effects of varying either the dielectric or magnetic re-laxation exponents ( and , respectively) in the upper layer(Figures 7c and 7d) are similar. These simulations correspond,respectively, to model sets 8 and 9 of Table 1, for which themain reflectivity is produced by the contrast, with small per-turbations caused by the relaxations. It must be emphasizedthat varying or results in frequency-dependent changesin both the real and imaginary parts of or , respectively(Figure 1).

Varying from 0 to 1 reduces the zero-offset reflection

amplitude by approximately 40%; varying over the samerange reduces it by only 15%. These increase to approximately60% and 25%, respectively, at larger offsets. Increases in re-flection coefficients associated with increasing contrasts areovershadowed by the decreasing reflection amplitudes causedby the accumulated attenuation along the propagation path.Note that this amplitude decrease with increasing is the re-verse of the direction in Figures 4d and 4e, in which reflectivityis produced by contrasts, but there is no attenuation betweenthe reflector and the free surface.

The fact that both these curve families are subparallel sug-gests that they provide small modifications to amplitudes thatare determined primarily by other variables, and that, in gen-

eral, they may be detectable, but not separable, in GPR data.The phase changes and attenuations associated with complex

FIG.5. Flattenedcommon-sourceGPR gathers formodels withvarious conductivities in the upper layer (model 6 of Table 1).(a) is for 0.0 S/m; (b) is for 0.01 S/m; (c) is for 0.05 S/m. Relativescaling factors are 1.0 in (a), 5.0 in (b), and 3000.0 in (c).

are largest when and are of similar magnitude and is close to 1 [equation (1) and Figure 1]. The same behav-ior is present in as it also is described by the Cole-Colemodel. As the dominant frequency of the source changes, rela-tive to the relaxation frequency ( or ), the dispersion andattenuation vary significantly [equation (1)]. Thus, the exam-ples above show specific instances that are representative, but

not comprehensive.

Responses as a function of frequency

Frequency dependence enters AVO computations in twoways: through the time-varying electrical properties [equa-tion (1) and Figure 1] and through the impedance [equa-tion (2)], which determines the reflectivity. It previously hasbeen shown (Turner and Siggins, 1994) that the reflection co-efficient at an interface between two media with constant tan (= / /) is frequency independent, but attenuationand dispersion of waves propagating through the media arefrequency dependent.

Figure 8 shows a representative example of reflection re-sponses and the corresponding AVO curves as a function offrequency for model 8 in Table 1 with a fixed distribution of di-electric relaxation ( = 0.7) in the upper layer. With all otherparameters fixed, reflected GPR waves are expected to show

FIG.6. Flattened common-source GPRgathersfor models withvarious relative magnetic permeabilities in the upper layer(model 7 of Table 1). (a) is for 1.1; (b) is for 1.9; (c) and (d)are for 2.8. In (d), of the lower layer is 7.5 rather than 20.5.F is the head wave generated at the free surface; B is the headwave generated at the reflector at the bottom of the layer. Bappears only when is sufficiently large in the upper layerthat its velocity becomes less than that in the lower layer. Therelative scaling factors are 1.0 for (a), (b), and (c), and 2.0 for(d).


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120 Zeng et al.

systematic changes in both attenuation and dispersion as thesource frequency changes [equations (1) and 5)]. Disper-sion is evident in Figures 8a8c; at fixed , velocity increases(e.g., the zero-offset time decreases) with increasing frequency(i.e., with decreasing ). Figure 8d contains a summary of theAVO behavior for this model. The migration of the maximumamplitude peak with frequency is a consequence of the corre-

sponding shift in critical angle as the effective refractive indexchanges with frequency (Figure 1).If finite was included in the model, there would also

be an attenuation with linear frequency dependence (TurnerandSiggins,1994) superimposedon thefrequency-symmetricalCole-Cole patterns associated with relaxations. This differencein the character of the frequency response potentially allowsseparation of the dielectric relaxation and conductivity effects;an example of a similar situation in the seismic context is givenby Kang and McMechan (1994). Thus, acquisition of multifre-quency GPR data is essential (but perhaps still not, for someparameters, sufficient) for resolving electrical properties in thesubsurface.

FIELD DATA EXAMPLE

The examples above were computed for flat-layered mod-els to avoid complications in data analysis associated with the

FIG. 7. (a) AVO curves for reflections from contrasts in (a) electrical conductivity , (b) relative magneticpermeability , (c) dielectric relaxation exponent , and (d) magnetic relaxation exponent . These correspondto models 6, 7, 8, and 9 of Table 1, respectively. The label on each curve gives the value of the property perturbedin the upper layer; quantities in (b), (c), and (d) are dimensionless, and in (a), is in S/m. For reference, the 0.0S/m curve in (a) is the same as the curve labeled 20.5 in Figure 4a. The curve labeled A in (b) is for the data inFigure 6d.

structure geometry. However, one of the main advantages ofthe finite-difference implementation is that it can handle ar-bitrary spatial distributions of electromagnetic properties andso is directly applicable to simulation of field GPR data. Re-flections from variably shaped reflectors will be modified fromthose shown above by lateral variations in velocity, reflectivity,and attenuation and by focusing and defocusing.

We considerexamples of two common-receiver gathers froma 100-MHz GPR profile collected in 1990 near Chalk River,Ontario, Canada. (We invoke reciprocity and model these ascommon-source gathers.) The site is in the Ottawa River val-ley in a composite fluvial and eolian environment that is idealfor GPR propagation. Auxiliary constraints and informationare available in the form of previous GPR profiles and coredboreholes.Detailedanalysesand geologic interpretation of thismultichannel data set are presented by Fisher et al. (1992) andGreaves et al. (1996). Modeling of constant-offset data froma different line in the same area was presented by Zeng et al.(1995), but the multichannel data have not been modeled pre-viously.

The two common-receiver gathers chosen for analysis are

from a portion of the GPR survey line that has reflectionsfrom a sequence of predominantly sand and gravel layers withmoderate structural dips (Figure 9). is assumed tobe 1 every-where because this is consistent with the geologicenvironment.


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The attenuation is low (see the inversion results of Cai andMcMechan [1999]) and, for the purpose of parameterization, itis assumed to be produced by alone. The rationale for this (intheabsence of independent measurements) is that even a smallconductivity will dominatethe attenuation in any medium (Fig-ure7), andthe frequency used (108 Hz) is farbelow thewater-relaxation frequency (1010 Hz), so is small. As discussedbelow, we expect a high porosity and high degree of satura-tion (with water near the freezing point), for which the effec-tive conductivity is very small. Thus, while the data do providegood examples of AVO behavior, they are lacking in some ofthe more dramatic aspects of attenuation and dispersion; thelatter will be left until a good relevant data set is found. The

FIG. 8. AVOresponses as a functionof frequency formodel 8 inTable 1, with = 0.7. (a), (b), and (c) are flattened reflectionsfor40, 125, and 400MHz, respectively. Alltraces have thesamescaling factor. The label on each AVO curve in (d) gives thedominant frequency of the source in MHz. For reference, the200-MHz curve would fall between the curves labeled 0.5 and0.8 in Figure 7c.

derived model is, of course, nonunique, but it is geologicallyplausible and is able to predict the main observations in thefield data (Figure 10). Ideally, the model building should bebased on lab measurements of the electrical properties, fromsamples of each of themain units, but such measurements werenot available.

The AVO features seen in the recorded data (Figures 10a

and 10c) have structural as well as lithologic contributions,and both are simulated by the finite-difference computations(Figures 10b and 10d). In both, the AVO of each reflector hasa pattern consisting of increasing amplitude, followed by de-creasing amplitude, as anticipated from the radiation-patterneffects described above. The position of the amplitude peak is

FIG. 9. Model for field data example. This model is derivedfrom the geometry in common-offset and migrated sections,velocity analyses from common-midpoint gathers, and litho-logic and depth control provided by a slightly off-line well corenear horizontal position 10 m. This auxiliary information is de-tailed in previous studies (Fisher et al., 1992; Greaves et al.,1996), and so is not shown here. In (d), points A and B corre-spond to the receiver locations for the common-receiver gath-ers in Figure 10. The numbered interfaces in (b) produce thecorrespondingly numbered waves in Figures 10b and 10d.


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at increasing offset with increasing time, consistent with a fixedcritical angle at the free surface.

Comparisons of real and predicted AVO curves for four in-dividual reflections in Figure 10 are presented in Figure 11.Although the main features and trends of the AVO behaviorsare simulated, the details and precise locationsof the maximumamplitudes are only approximate. Further refinements may im-

prove the fit, but the present model is already the result of 135iterations. We attribute the remaining misfit primarily to thefact that it is not possible to simulate exactly the response of a3-D structure with a 2.5-D model.

The polarity of the reflections is consistent with the litho-logic predictions. Reflections from the sequence of sands aregenerally of negative polarity, corresponding to increasing per-mittivity (decreasing velocity and decreasing impedance) withdepth. The reflection from the top of a clay layer (the deep-est modeled reflector, at about 16-m depth) has positive po-larity corresponding to a decreasing permittivity (increasingvelocity and increasing impedance) with depth. Low poros-ity and, hence, a small water content in the clay explain theobserved character of the reflection. In this low-conductivity

(cold-water-saturated) environment, changes dominate thereflectivity (Figures 9b and 9c). The unusually high values of in some parts of the model are consistent with previous esti-mates of water content (using this same data set) by Greaveset al. (1996). Both their and our results for these high valuesare consistent with fully water-saturated porosity in the rangeof 40% by volume (Topp et al., 1980).

FIG. 10. Field (a and c) and corresponding synthetic (b and d) common-receiver gathers. (a) and (b) are for areceiver at point A in Figure 9d; (c) and (d) are for point B. Numbered reflections in (b) and (d) are producedby the similarly numbered model elements in Figure 9b; 14 is the air wave, 12 and 13 are a complex interferencebetween the direct ground wave and reflections from shallow structure, and the other waves are reflections anddiffractions.

DISCUSSION AND CONCLUSIONS

We have presented a suite of synthetic AVO computationsby systematically varying each of the electromagnetic proper-ties of a layer-over-a-half-space model. The synthetic GPR re-sponses include all the antenna radiation-pattern, free-surface,and propagation (attenuation and dispersion) effects, not just

the reflection coefficients. Thus, the simulated wide-aperturegathers contain AVO behaviors that are comparable to fieldGPR data. The main features in two representative fieldcommon-receiver gathers are predicted fairly well (Figures 10and 11) by synthetic data computed for a model (Figure 9)whose average parameters were estimated independently. Thenew model involves refinement through the addition of detailsto the average model, so the two models are compatible witheach other.

Although only simple models were considered in this pa-per, an important advantage of the finite-difference approachis that AVO curves can be predicted for any model with ar-bitrary complexity, and for any frequency. Thus, the tools andmethodology illustrated here can be used in a broader context

of GPR data prediction, field-survey design, and data analy-sis and interpretation (e.g., Xu and McMechan, 1997). In thelonger term, the finite-difference solution may set the stagefor full-wave multifrequency inversion of GPR data. From theabove results, it is expected that not all parameters will beresolvable. For example, contrasts in dielectric and magneticrelaxations produce very similar AVO behaviors (Figures 7c


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and 7d) and have relatively small reflectivity compared withthose produced by independent changes in , , and , andthus may not be separable in typical real GPR measurements.However, use of additional independent data from multiplefrequencies, fromlab measurements(e.g.,Olhoeft and Capron,1993), and/or from other antenna polarizations (e.g., Roberts,1994) may provide tractable solutions.

The shape of the AVO responses of all the models consid-ered is dominated by the amplitude peak associated with thecritical angle at thefreesurface. The positionof this peak is con-trolled primarily by the dielectric permittivity of the materialimmediately beneath the antennas (Figure 4a). The position ofthe peak and the rates of amplitude decay to either side of thepeak (which can be substantially different) depend on whichelectromagnetic properties change at the target interface andby how much (Figures 4 and 7).

Some of the results we have obtained (especially the suitesof curves in Figures 4 and 7) potentially can be used for em-pirical characterization and interpretation of targets in fielddata. For example, the overall shapes and rates of amplitudedecay of AVO curves for magnetic targets (Figures 4c and 4d)

are diagnostically different from those for conductive targets(Figures 4b and 4d).

One of the difficulties in interpreting AVO curves (which isbeyond the scope of this paper) is that models relating petro-physical properties to electrical properties are still incomplete.Therefore, although some attempts appear to be qualitativelysuccessful (Hoekstraand Delaney, 1974; Wobschall,1977; Topp

FIG. 11. Comparison of field (solid line) and synthetic (dashed line) AVO data. (a) and (b) are for the reflectionslabeled 8 and 1, respectively, in Figure 10b. (c) and (d) are for the reflections labeled 3 and 1, respectively, inFigure 10d. The synthetic data are scaled to have amplitudes similar to the field data, because we are concernedonly with relative amplitudes.

et al., 1980; Feng and Sen, 1985; Hallikainen et al., 1985;Redman et al., 1994; Olhoeft and Capron, 1994; Greaves et al.,1996), factors such as grain contact and fluid/solid interfacephenomena and electrochemical processes at the microscaleremain elusive (Olhoeft, 1987; Gueguen and Palciauskas,1994).

Factors that have not been considered here but that also will

influence AVO behavior include the 3-D reflector geometry,scattering, anisotropy, and lateral variations in the overbur-den. All the models are 2.5-D, so off-line 3-D effects were notincluded. Only one antenna polarization for one antenna typewas considered. Thus, the present paper is only a demonstra-tion of feasibility, not an exhaustive investigation. It is clear,however, that AVO measurements may provide constraints onthe composition of a target through more comprehensive in-terpretation of the GPR wavefield.

For the modeling, we used an effective medium represen-tation based on the empirical Cole and Cole (1941) model,which is deemed adequate for practical model descriptionwhen the GPR wavelengths are large compared with the grainsize (Gueguen and Palciauskas, 1994). This representation is

flexible in terms of its ability to mimic thecomplexresponses ofreal materials at the macroscale, and the required parameterscan be obtained by lab measurements. However, because thisrepresentation is empirical, it does not directly produce uniqueinterpretations in terms of micromechanisms and processes.

For typical situationsin realearth materials, reflectionampli-tudes are most influenced, in order of importance, by contrasts


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(when they are present) in dielectric permittivity, electricalconductivity, magnetic permeability, dielectric relaxation, andmagnetic relaxation (Figure5). Attenuation is mostdependent,in order of importance, on electrical conductivity, dielectric re-laxation, and magnetic relaxation (Figure 7). Decreasing con-ductivity, decreasing permittivity, or increasing permeability,all increase the electromagnetic impedance (equation 2), with

corresponding changes in AVO behavior.

ACKNOWLEDGMENTS

The research leading to this paper was supported by the U.S.Department of Energy under Contract DE-FG03-96ER14596,by the Texas Advanced Technology Program under grant09741-035, and by the Sponsors of the UT-Dallas GPR Con-sortium. Constructive criticisms by David Alumbaugh, KarlEllefsen, and two anonymous reviewers are much appreciated.The field data in Figure 11 were recorded in 1990 with field sup-port provided by Sensors & Software, Inc. This paper is Con-tribution No. 908 from the Geosciences Department at TheUniversity of Texas at Dallas.

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