frtimothy c. johnson, partha s. routh and michael d ... · geophys. j. int. (2005) 162, 9–24 doi:...

Geophys. J. Int. (2005) 162, 9–24 doi: 10.1111/j.1365-246X.2005.02658.x

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Fresnel volume georadar attenuation-difference tomography

Timothy C. Johnson, Partha S. Routh and Michael D. KnollCenter for Geophysical Investigation of the Shallow Subsurface (CGISS), Department of Geosciences, Boise State University, 1910 University Drive, Boise,ID 83725 USA. E-mail: [email protected]

Accepted 2005 April 11. Received 2005 February 25; in original form 2004 November 5

S U M M A R YGeoradar attenuation-difference tomography is a useful tool for imaging temporal and spatialchanges in bulk electrical conductivity due to fluid flow and other subsurface processes. Themost common method of attenuation-difference tomography employs the ray approximationwhere waves are assumed to propagate at infinite frequency. Ray approximation causes sig-nificant model error that generates artefacts and loss of resolution in tomographic images.In this paper we propose an efficient method of computing Fresnel volume sensitivities usingscattering theory. These sensitivities account for finite frequency propagation and represent thephysics of electromagnetic propagation more accurately than ray theory. As expected, Fresnelvolume sensitivities provide better data prediction than ray-based sensitivities. We use singularvalue decomposition analysis to show how and why this physical improvement allows Fresnelvolume inversions to recover localized targets and resolve bulk conductivity changes betterthan ray-based inversions. The model basis functions and singular values corresponding tothe scattering theory sensitivity kernel are similar to the exact full-waveform basis functionsand singular values. The similarity of the basis functions and singular values suggests thatthe scattering theory inverse estimates are close to the full-waveform estimates, but they areobtained with a significant decrease in computational effort.

Key words: attenuation, electromagnetic modelling, Frechet derivatives, inversion,scattering, tomography.

1 I N T RO D U C T I O N

Characterization of the aquifer properties that control flow and trans-port at scales typical of environmental and engineering problemsis an area of active research. Researchers are developing innova-tive methods of utilizing geophysical surveys to provide additionalinformation concerning aquifer properties and to help constrainthe range of possibilities in hydrogeological parameter estimates(Hyndman et al. 1994; Copty & Rubin 1995; Hubbard & Rubin2000; Day-Lewis et al. 2002; Barrash et al. 2003; Goldstein 2004).Radar attenuation-difference tomography can be used to image sub-metre spatial and temporal changes in bulk electrical conductivityinduced by the movement of a conductively anomalous fluid suchas a tracer or contaminant. Such transport information is typicallysensitive to important hydrogeological properties such as porosityand hydraulic conductivity and can provide valuable informationconcerning the distribution of those properties.

A number of radar attenuation-difference surveys have been con-ducted in near-surface environments. In a series of experiments,Lane et al. (1999) showed how cross-well attenuation-differencedata can be used to image fractures in crystalline bedrock con-taining saline tracer. In a follow-on study Day-Lewis et al. (2002)developed a novel approach to time-lapse inversion that accountsfor the movement of the tracer over the course of the data collec-tion time to improve image resolution. As with most applicationsof radar tomography, ray theory was used to represent the physics

of electromagnetic (EM) wave propagation underlying the inversereconstruction of the tracer distribution in each of these cases. Themajority of tomographic inversions for velocity or attenuation struc-ture (in both seismic and radar applications) are based on ray theoryprimarily because ray theory is well understood and computationallyefficient allowing inversions of large data sets.

In ray theory, the sensitivity of pulse traveltime or attenuation tothe electromagnetic properties of the medium is reduced to a lineintegral along the travel path between the source and the receiver.Ray-geometrical representation of wave propagation is a mathemat-ical construct that is valid in the limiting case of infinite-frequencypropagation. In finite-frequency propagation the velocities and am-plitudes of waves are sensitive to some volume surrounding thefastest travel path. These sensitivity volumes were first describedin terms of seismic waves by Hagedoorn (1954) and later definedas Fresnel volumes by Kravstov & Orlov (1980). Collapsing theseFresnel volumes of sensitivity to 1-D curves along the fastest travelpath has several limitations. In velocity tomography, for exam-ple, rays are deflected around low-velocity regions such that high-velocity regions are preferentially sampled biasing the solutiontoward high velocities (Wielandt 1987; Nolet 1987). Ray theory doesnot account for many low-velocity regions adjacent to the fastesttravel path to which corresponding traveltimes may be sensitive.

Several techniques have been developed to estimate the spatialdistribution of the sensitivity of seismic arrival time to seismicvelocity distributions, which are often referred to as wave path

C© 2005 RAS 9

10 T. C. Johnson, P. S. Routh and M. D. Knoll

methods (Luo & Schuster 1991; Woodward 1992; Stark & Niko-layev 1993; Vasco & Majer 1993; Vasco et al. 1995). Generally, thefull acoustic-wave equation is used to derive the wave path sensitiv-ity distribution associated with a particular source–receiver pair andvelocity structure (Vasco & Majer 1993; Spetzler & Snieder 2004).Because they are based on a finite-frequency wave equation, thewave path sensitivities are physically more accurate than rays andinclude the effects of finite-frequency propagation that are neglectedin ray theory. The benefits of accounting for finite-frequency effectsare numerous, including the ability to image smaller-scale featuresand reduce tomographic artefacts. The disadvantage of wave pathtomography is computational cost, since each wave path calculationrequires a forward and backward propagation of a full wavefield andresidual or adjoint wavefield, respectively.

In this paper we present an efficient method of computing thesensitivity of finite-frequency radar attenuation data to changes inbulk conductivity. We begin with a review of ray-based attenuation-difference tomography. Then we present the Fresnel volume sensitiv-ity computation by defining the Fresnel volume to which attenuationdata are sensitive, and then using scattering theory to compute thesensitivities within the volume. We compare the Fresnel volume sen-sitivity distributions to those derived by perturbing a full-waveformfinite-difference solution (Holliger & Bergmann 1999) and showhow the scattering theory sensitivities are more accurate than theray-based sensitivities with a data prediction example. Followingthis discussion we use singular value decomposition (SVD) analysisof a synthetic example to show how this physical improvement pro-vides better target localization in inverse estimates, giving markedimprovements in tomographic resolution over ray-based inversions.By decomposing the sensitivity kernels using SVD we analyse thenature and scaling of the basis functions used to construct the in-verse estimates. The slowly varying and localized composition of thescattering theory basis functions allows Fresnel volume inversions toresolve localized targets more accurately than ray-based inversions.In addition, the full-waveform and scattering theory basis functionsand singular values display a similar character in both shape andmagnitude, reinforcing our intuition about the validity of Fresnel-based sensitivities. The similarity between the scattering theory andfull-waveform basis functions and singular values suggests that thescattering theory solution is similar to the full-waveform solution,but obtained with a significant decrease in computational effort.

2 T H E O R E T I C A L D E V E L O P M E N T

In this section we briefly review the ray theoretical approach to re-late amplitude data to attenuation changes or perturbations in themedium. Then we discuss the Fresnel theory that relates changesin amplitudes to changes in bulk electrical conductivity within theFresnel volume. Finally we present a method for computing thesensitivity of amplitudes to changes in the bulk conductivity of dis-cretized cells within the Fresnel volume.

2.1 Ray theory overview

The electric field recorded at the receiver at time ta corresponding tosource–receiver configuration i is denoted by ea,i . The ray-geometricfar-field expression describing the amplitude (Aa,i ) of ea,i is givenby (Holliger et al. 2001; Holliger & Bergmann 2002; Day-Lewiset al. 2002)

Aa,i = A0a,i�sa(φi )�ra(φi )exp

(− ∫

l αa(s) ds)

∫l dsa

(1)

where l is the length of the ray path from source to receiver, A0a,i

is the effective amplitude of the source at the position related to iand time ta, αa is the attenuation coefficient distribution at time ta,�sa is the source radiation pattern at time ta, �ra is the receiverradiation pattern at time ta and φ i is the angle between source andreceiver with respect to horizontal. If radiation patterns and sourceamplitudes are time invariant the difference between ln(Ab,i ) andln(Ab,i ) for a discretized medium can be expressed as

ln(Aa,i ) − ln(Ab,i ) =N∑

j=1

�α j�ri j (2)

where �α j = αb, j − αa, j , �rij = �r a,i j = �r b,i j and N is the num-ber of cells. Eq. (2) provides a linear mapping relating the changesin attenuation between times ta and tb to the corresponding changein amplitude of the data. In matrix notation

δD = J S�α (3)

where δD = ln(Aa,i ) − ln(Ab,i ) and J S = �ri j . Eq. (3) is theforward model for ray-based attenuation-difference tomography. Inthis formulation radiation pattern effects cancel by differencing thelog amplitudes if the radiation patterns are time invariant. This isan attractive feature of attenuation-difference tomography becauseradiation patterns are typically difficult to determine. A subtle butimportant caveat to this result is that it was derived under the ray-geometric approximation and may not apply to the finite-frequencypropagation case. Later we show that attenuation-difference data areindeed sensitive to radiation patterns in finite-frequency propaga-tion whether radiation patterns are time invariant or not. The Fresnelzone sensitivity distributions we present here do account for radia-tion pattern effects assuming they are known or can be sufficientlyapproximated.

As a final note, some authors (e.g. Lane et al. 1999; Peterson2001) derive eqs (1) and (2) slightly differently, depending primar-ily on the way in which Aa,i is defined. The differences arise fromthe ray-geometric approximation itself which requires all of the sen-sitivity assigned to Aa,i to lie upon a curve regardless of the timewindow within which Aa,i is defined. Thus the definition of Aa,i

is somewhat arbitrary. In finite-frequency propagation the region inspace to which Aa,i is sensitive depends on how Aa,i is defined.For instance, if Aa,i is defined as the peak of the first pulse, theregion in space to which Aa,i is sensitive includes all points suchthat energy scattered from those points contributes to the first pulse.In the following discussion, we define Aa,i to be sensitive to onlythe first Fresnel volume characterized by the period of the first halfcycle of ea,i and the propagation velocity. Then we use scatteringtheory to estimate the sensitivity of amplitudes to bulk conductiv-ity changes within the Fresnel volume assuming constant velocitypropagation.

2.2 Fresnel volumes, attenuation-difference dataand the sensitivity kernel

This section describes how we define attenuation-difference data andhow those data are related to the first Fresnel volume characterizedby the period of the first half cycle (or first pulse) and velocity ofthe recorded wavelet. The definition of the first Fresnel volume ofconstructive interference, or simply Fresnel volume, was originallygiven by Kravstov & Orlov (1980) and is illustrated in Fig. 1. TheFresnel volume is the region defined by

ts j + tr j − tsr ≤ T/2 (4)

C© 2005 RAS, GJI, 162, 9–24

Georadar attenuation-difference tomography 11

Figure 1. (a) Field examples of ea,i and eb,i gathered during a conductivetracer experiment and the corresponding theoretic Fresnel volume boundary.Note t sr ≈ 81.5 ns and T /2 ≈ 12.3 ns are approximately equal for each traceand that the region between t sr and t fp includes the ‘first pulse’. The velocitywas estimated by dividing the offset by the time to the peak of the first pulsewhich is most appropriate for Fresnel zone applications (Vasco et al. 1995).(b) The Fresnel volume boundary bounds the region in space to which thefirst pulse is sensitive.

where t s j is the traveltime from the source to some scattering pointin space (cell j in the context of this paper), t r j is the traveltimefrom the same point to the receiver, t sr is the traveltime from sourceto receiver and T is the period of the propagating wave. Energy inthe first T /2 seconds of the wavelet is only sensitive to points lo-cated within the first Fresnel volume because energy scattered frompoints outside of this volume arrives after time t sr +T /2 (Cerveny& Soares 1992). We define T /2 as the length of time between thefirst arrival (t sr) and the end of the first pulse (t fp) of the arrivingwavelet so that the first pulse is only sensitive to the Fresnel vol-ume defined by T . Higher-order Fresnel zones are also sensitive toscattered energy, but computing the sensitivities for each would re-quire an excessive computational burden and approaches the realmof full-waveform inversion where more efficient techniques (suchas adjoint sensitivities) are available.

For a constant-velocity medium the Fresnel volume assumes anellipsoidal shape with the focus points at the source and receiverpositions as shown in Fig. 1. As frequency decreases the periodincreases and the Fresnel zone widens indicating that the energycontained in the first pulse is sensitive to a wider region in space. Asfrequency increases towards infinity the period approaches zero andthe Fresnel zone becomes very narrow, indicating that the first pulseis sensitive only to the locus of points lying on a curve between thesource and receiver as in the ray approximation. As noted earlier,we assume a constant-velocity medium so that rays are straight andFresnel volumes are ellipsoidal. The straight-ray approximation isvalid in many saturated shallow subsurface environments becausevelocity contrasts tend to be weak and source–receiver distancesare small which limits ray bending (Lane et al. 1999; Day-Lewiset al. 2002; Buursink 2004). However, Fresnel volumes can alsobe computed for media with heterogeneous velocity (Spetzler &Snieder 2004).

To introduce our definition of attenuation-difference data, sup-pose radar data are collected between two boreholes at some back-

ground time ta. The time domain trace of the vertical component ofthe electric field (E z) corresponding to the source–receiver orienta-tion i is given by ea,i . A trace with the same acquisition geometrycollected at some later time tb when an electrically anomalous fluidhas invaded the region between the source and receiver, is labelledeb,i . If the fluid is conductive, the amplitude of ea,i will be greaterthan the amplitude of eb,i because EM wave attenuation increaseswith conductivity. In low-loss conditions, the relationship betweenthe natural logarithm of EM amplitude and bulk conductivity is ap-proximately linear (see Appendix A). With this in mind, we definethe attenuation-difference data as

δDi = ln

(∫ tfp

tsr

e2b,i (t) dt

)− ln

(∫ tfp

tsr

e2a,i (t) dt

)(5)

where t sr is the first arrival time for traces ea,i and eb,i , and t fp

is the time to the end of the first pulse or the first zero crossingafter t sr. We assume that t sr is equal for each trace. This assump-tion is valid when the change in bulk conductivity between timesta and tb is not extreme, because EM wave velocity is a weak func-tion of bulk conductivity (Jackson 1999). Eq. (5) also assumes t fp

is constant between traces which is approximately true in the ab-sence of a significant change in dispersion from ea,i to eb,i . In otherwords, the conductivity change between times ta and tb does notsignificantly affect the EM wave velocity. The kinematic parts ofthe wave equation are equal for both times; only wave amplitudesare affected. Under this assumption, the integration limits in eq. (5)are equal for each trace. Field examples of ea,i and eb,i along withthe corresponding theoretical Fresnel zone boundary (?) are shownin Fig. 1.

To construct the forward model, let the conductivity distributionat times ta and tb be denoted by σ a(r) and σ b(r), respectively, wherer is the position vector. Eq. (5) then becomes

δDi = ln

(∫ tfp

tsr

e2i (σb(r), t) dt

)− ln

(∫ tfp

tsr

e2i (σa(r), t) dt

). (6)

Let δσ (r) represent a change in the conductivity field about σ a(r)such that σ b(r) = σ a(r) + δσ (r). Then

δDi = ln

( ∫ tfp

tsr

e2i (σa(r) + δσ (r), t) dt

)

− ln

( ∫ tfp

tsr

e2i (σa(r), t) dt

). (7)

If we represent the integrals by

Fi (σ (r)) =∫ tfp

tsr

e2i (σ (r), t) dt, (8)

then eq. (7) can be rewritten and expanded in Taylor series to obtain

δDi = ln

(Fi (σa(r)) +

⟨∂ Fi (σa(r))

∂σa(r), δσ (r)

⟩+ O

( ‖δσ (r)‖2))

− ln [Fi (σa(r))] (9)

where 〈·, ·〉 denotes the inner product in the spatial domain and∂ Fi(σ a(r))/∂σ a(r) are Frechet derivatives. When δσ (r) is small,second and higher-order terms of the expansion are insignificantand eq. (9) reduces to

δDi =⟨

1

Fi (σa(r))

∂ Fi (σa(r))

∂σa(r), δσ (r)

⟩. (10)

Next we express the conductivity change in discrete form as

δσ (r) =N∑

j=1

δσ j� j (r) (11)

C© 2005 RAS, GJI, 162, 9–24


where δσ j is the conductivity change in cell j, N is the number ofcells,

� j (r) ={

1 if r ∈ v j

0 otherwise(12)

is the basis function used to discretize the model and vj is the volumeto which δσ j applies. Substituting this discretization into eq. (10)and taking the summation outside of the inner product gives

δDi =N∑

j=1

J Fi jδσ j (13)

where J Fi j = ∂ln [Fi(σ a(r))]/∂σ j . Eq. (13) is the discretized for-

ward model for Fresnel volume attenuation-difference tomography.Errors in the physics represented by J F arise primarily due to thetruncation of the Taylor series expansion in eq. (9) which neglectsthe effects of multiple scattering. This effect is evident in the ex-pression for J F

i j which is the sensitivity of the log first pulse energyof ei(σ a(r)) to a single scatterer δσ j .

We compute J Fi j with the forward finite-difference operator given

by

J Fi j =

ln(∫ tfp

tsre2

i

(σa(r) + δσ j , t

)dt

)− ln

(∫ tfptsr

e2i (σa(r), t) dt

)δσ j

.

(14)

For notational convenience we drop the time dependence in ei. Wesolve for ei (σ a(r)) and ei (σ a(r) + δσ j ) using scattering theory andnumerically compute the integrations in eq. (14). The backgroundfield ei (σ a(r)) is computed assuming constant σ a(r) and ei(σ a(r) +δσ j ) is computed using the Born approximation. Next we progressthrough the theory and mechanics of computing J F

i j , beginning withthe integral solutions for ei(σ a(r)) and ei(σ a(r) + δσ j ) includingradiation pattern effects.

2.3 Fresnel volume sensitivities: theory

In this section we develop the equations describing the backgroundand scattered fields. Notation for this development is shown in Fig. 2.We present the solution to the Helmholtz equation for a point sourcedelta function and a small conductive scatterer, beginning with thefrequency domain versions of Ampere’s law and Faraday’s law wherethe source current density is included in the J(r, ω) term. This isgiven by

∇ × E(r, ω) = −iωB(r, ω) (15)

∇ × H(r, ω) = J(r, ω) + iωD(r,ω) (16)

Figure 2. Diagram showing notation for the theory.

with the non-dispersive constitutive relations given by

J(r, ω) = σ (r)E(r, ω) (17)

D(r, ω) = ε(r)E(r, ω) (18)

B(r, ω) = µ(r)H(r, ω). (19)

Here we assume a time harmonicity of eiωt where ω is the angularfrequency and t is the time. E is the electric field strength, B is themagnetic induction, H is the magnetic field strength, J is the galvaniccurrent density, D is the displacement current, σ (r) is the electricalconductivity, ε(r) is the dielectric permittivity, µ(r) is the magneticpermeability and r is the position vector with respect to the systemorigin. We assume that variations in ε may be considered negligi-ble throughout the domain and that µ may be considered constantand equal to its free space value µ0. The propagation velocity ofelectromagnetic waves is a strong function of the dielectric permit-tivity. If variations in ε are small, velocity variations in the mediumare also small, and ray representations of the wave propagation areapproximately straight.

Substituting eq. (19), into eq. (15) and taking the curl of bothsides gives

∇(∇ · E(r,ω)) − ∇2E(r, ω) = −iωµ0∇ × H(r, ω). (20)

Under low-loss conditions the conduction currents are insignificantin comparison to the displacement currents (Jackson 1999) so that∇ ·E (r, ω) is negligible and eq. (20) reduces to

∇2E(r, ω) = iωµ∇ × H(r, ω). (21)

Next we substitute eqs (16) and (18) into eq. (21) to get

∇2E(r, ω) = iωµ(J(r, ω) + iωεE(r, ω)). (22)

We separate the current density into two terms, one for the currentdistribution in the source antenna, (S(r, ω)), and one for the currentdistribution in the medium as described by eq. (17)

J(r, ω) = S(r, ω) + σ (r)E(r, ω). (23)

The conductivity distribution is described by

σ (r) = σa(r) + δσ (r) (24)

where σ a(r) is the background conductivity and δσ (r) representsa change in the conductivity from the background. Substitutingeqs (23) and (24) into eq. (22) and rearranging terms gives

∇2E(r, ω) + k2E(r, ω) = iωµ0 S(r, ω) + iωµ0δσ (r)E(r, ω) (25)

where k = √ω2µ0ε − iωµ0σa(r) is the complex wavenumber.

Eq. (25) is the Helmholtz equation for the electric field due to twocurrent sources. The first current source is S(r, ω), which is thecurrent produced in the source antenna. The second source is δσ (r)E(r, ω), which is the scattering current produced by the change inconductivity represented by δσ (r). That is, the scatterer acts as aneffective current source whose strength is proportional to the con-ductivity perturbation δσ (r) and the electric field strength E(r, ω)within the scattering volume.

2.4 Background field solution

Since we are only solving for the z dimension of the electric field,we drop the vector notation and note that all fields and sources aregiven with respect to the z dimension for the rest of the derivation. Wefind the solution to eq. (25) by decomposing the total field into thepart caused by the background conductivity σ a(r) which we denote

C© 2005 RAS, GJI, 162, 9–24


E 0(r, ω), and the part caused by the scatterer which we denoteE 1(r, ω). The total field is given by E(r, ω) = E 0(r, ω) + E 1(r, ω).The Helmholtz equation for E 0(r, ω) is given by

∇2 E0(r, ω) + k2 E0(r, ω) = iωµ0 S(r, ω) (26)

with the boundary conditions that E 0 (r, ω) → 0 as r → ∞. Tofind the solution to eq. (26) we assume a homogeneous backgroundconductivity and a boundless domain, then employ the full-spaceGreen’s function for the field at r given a source region r′. In threedimensions the Green’s function is (Ward & Hohmann 1988)

G(r, r′, ω) = 1

4π |r − r′| e−ik|r−r′ |. (27)

In two dimensions the Green’s function is

G(r, r′,ω) = i

4H (1)

0 (k|r − r′|) (28)

where H (1)0 (k|r − r′ |) is the Hankel function (Abramowitz & Stegun

1975). The whole space Green’s function is appropriate for our prob-lem because both the source and receiver are within the medium. IfS is a point source located at r0 and S(ω) is the frequency domainrepresentation of the source wavelet, then S(r0, ω) = δ(r0)S(ω), andthe integral solution reduces to

E0(r, ω) = iωµ0

4π

1

|r − r0| S(ω)e−ik|r−r0| (29)

in three dimensions and

E0(r, ω) = −ωµ0

4H (1)

0 (k|r − r0|)S(ω) (30)

in two dimensions.

2.5 Scattered field solution

The equation for the scattered field E 1 (r, ω) caused by a conductivescatterer at V ′ is given by

∇2 E1(r, ω) + k2 E1(r,ω) = iωµ0δσ (r)E(r, ω) (31)

which has the integral solution

E1(r, ω) = iωµ0δσ j

∫v j

G(r, r j , ω)E(r j , ω) dv j . (32)

using the discretization given in eq. (11). We assume that vj is smallenough that the Green’s function is approximately constant within vj

at a particular frequency, and δσ j may be taken out of the integral.We also assume vj is small enough that E(r j , ω) is approximatelyconstant within vj and may also be taken out of the integral. Thisassumption requires that the dimensions of vj be much smaller thanthe wavelength of E (r j ,ω) (Ishimaru 1978). Under these conditionseq. (32) reduces to

E1(r, ω) = iωµ0δσ jvG(r, r j , ω)E(r j , ω) (33)

where r j denotes the position of the centroid of vj for the remainderof the derivation.

To obtain the scattered field solution in eq. (33) we replaceE(r j , ω) with the background field solution assuming that E(r j ,ω) ≈E 0 (r j , ω), or apply the Born approximation (Ishimaru 1978).After substitutions this gives

E1(r, ω) = − ω2µ20v jδσ j

16π2|r − r j ||r j − r0| e−ik(|r−r j |+|r j −r0|) S(ω). (34)

in three dimensions.

The total electric field measured at the receiver is

E(r, ω) = iωµ0

4π

1

|r − r0| S(ω)e−ik|r−r0|

− ω2µ20v jδσ j

16π 2|r − r j ||r j − r0| e−ik(|r−r j |+|r j −r0|) S(ω). (35)

Eq. (35) is the fundamental equation used in computing the EM fielddue to a conductive perturbation in a homogeneous medium. It isthe total field solution for a single scatterer under the Born approx-imation. Note that for scatterers located along the ray, the complexexponential terms of the background and scattered fields are equiv-alent. However, the background field term contains the complexnumber i indicating that the background field is phase shifted by90◦ with respect to the scattered field for scatterers along the ray.This becomes an important factor when describing the propertiesof the sensitivity distributions shown in the results section. In thenext section we show how this equation is used to approximate theFresnel volume sensitivities.

3 S E N S I T I V I T Y C O M P U TAT I O NA N D A N A LY S I S

3.1 Sensitivities using scattering theory

We begin by discretizing the 3-D computational domain into smallcells. The volume vj of cell j must be small enough that the electricfield and the Green’s function within the cell are approximatelyconstant at any moment in time so that eq. (34) holds. We show inSection 4.1 that for a 100 MHz Ricker wavelet with a correspondingdominant wavelength of approximately 1 m, the integral solutionpresented here matches well with the finite-difference solution whenthe cells are cubes with 0.25 m sides, or about 25 per cent of thedominant wavelength. Once a discretization is chosen, the centrepoint of cell j is used to compute |r j − r0| and |r − r j | in thesolution for the scattered field caused by cell j.

The time domain source term s(t) is a shifted Ricker waveletdefined by

s(t) = [1 − 2π 2 f 2

0 (t − ts)2]

exp[ − π2 f 2

0 (t − ts)2]

(36)

where f 0 is the dominant source frequency and t s is a time shift toensure causality. Once the source function is defined, E0(r, ω) iscomputed by eq. (29) and ei(σ a(r)) is computed by

ei (σa(r)) = F−1 (E0(r, ω)) (37)

where F−1() is the time-frequency inverse Fourier transform.Eq. (37) is the background field solution for a point source with

an isotropic radiation pattern. To illustrate the implementation ofmore complicated radiation patterns, consider the general sourceand receiver radiation patterns �s(φ i ) and �r(φ i ) where φ i is thesource–receiver angle with respect to horizontal associated withthe ith source–receiver configuration. Then the background field isgiven by

ei (σa(r)) = F−1 (E0(r, ω)) �s(φi )�r(φi ). (38)

In this paper we assume dipole-type radiation patterns such that�s (φ i ) = �r(φ i ) = cos (φ i ). The final step with respect to thebackground field is to integrate [ei(σ a (r))]2 over the first pulse. Wedo this with a simple algorithm that identifies the arrival time t sp andthe end of the first pulse t fp, as shown in Fig. 3, and then integrates[ei(σ a (r))]2 numerically between these limits.

C© 2005 RAS, GJI, 162, 9–24


Figure 3. Integration limits of background wavelets computed with thescattering theory solution. The algorithm determines t fp by finding the zerocrossing of the first pulse. The first arrival time t sp is determined by pickingthe time where the wavelet amplitude exceeds a user-defined value. The firstpulse time T /2 is slightly greater than 4 ns in this case. In practice thiswavelet would be used to compute the Fresnel volume sensitivities of a fieldtrace with a first pulse time of approximately 4 ns.

For those cells within the Fresnel volume corresponding to shotreceiver orientation i, the total field for each scatterer ei(σ a(r) +δσ j ) is computed by adding the scattered field solution to the back-ground field solution. The scattered field solution, which we labele1i (σ a(r) + δσ j ), is computed in a manner analogous to the com-putation for ei(σ a(r)). First E 1(r, ω) is computed by eq. (34). Forisotropic radiation and sensitivity pattern, e1i (σ a(r) + δσ j ) is givenby

e1i (σa(r) + δσ j ) = F−1 (E1(r, ω)) . (39)

If we assume isotropic scattering and source and receiver radiationpatterns given by �s(φ si j ) and �r(φ ri j ), then e1i j (σ a(r) + δσ j ) isgiven by

e1i (σa(r) + δσ j ) = F−1 (E1(r, ω)) �s(φsi j )�r(φri j ) (40)

where φ si j and φ ri j are the vertical angles between cell j and thesource and receiver respectively for source–receiver configurationi. Eq. (40) explicitly shows how the amplitude of the scattered fieldis affected by the radiation patterns and the position of the scattererwith respect to the source and receiver. The total field is given by

ei (σa(r) + δσ j ) = ei (σa(r)) + e1i (σa(r) + δσ j ). (41)

Finally, we integrate over the first pulse of ei(σ a(r) + δσ j )2 asdiscussed above and compute J F

ij as shown in eq. (14). A similarprocedure can be used to compute J F for the 2-D problem using the2-D Green’s function shown in eq. (28) if the source is 2-D. Becausewe are using a 3-D source function, we compute the 2-D sensitivitiesby integrating the 3-D J F along the strike dimension.

3.2 Finite-difference approach

We verify the analytical solutions for ei(σ a(r)), ei(σ a(r) + δσ j )and e1i (σ a(r) + δσ j ) by comparison with the 2-D finite-differencetime-domain solution to the vertical component of the EM fieldgiven by Holliger & Bergmann (2002). They also assume dipole-type antennas with a time-shifted Ricker source wavelet so the re-sults are directly comparable. The finite-difference solutions for the

background and total fields are computed directly. Then the scat-tered field is determined by e1i (σ a(r) + δσ j ) = ei(σ a(r) + δσ j ) −ei(σ a(r)) for comparison with the scattering theory solution. Wealso use the finite-difference solution to produce synthetic data tocompare with the Fresnel zone and straight-ray forward and inversesolutions. Finally, we compute the full-waveform sensitivity distri-bution about the synthetic solution in a manner analogous to theFresnel zone computations. However, in this case σ a (r) representsthe true conductivity distribution so that the sensitivities are com-puted about the solution. We denote the full-waveform sensitivitydistribution by J FW and use J FW as the benchmark solution to com-pare the properties of J S and J F in the SVD analysis.

3.3 Synthetic data inversion and SVD analysis

One of the goals of this paper is to show quantitatively how theFresnel zone sensitivity kernel J F represents the physics of wavepropagation better than the ray theory sensitivity kernel J S, andto show how inverse solutions produced using J F result in better-resolved images with fewer artefacts. We do this for a synthetic testcase by comparing the properties of J FW with J F and J S using SVDanalysis. We consider the case where there are more parameters thandata and regularization is introduced only by truncating the numberof basis functions in the solution obtained from the SVD analysis. Noadditional a priori constraints are imposed in the inverse solution.To make the analysis directly comparable, we produce synthetic dataunder low-loss conditions so that the relationship between attenua-tion coefficient and bulk conductivity is approximately linear, andthe straight-ray sensitivity kernel may be given as (see Appendix A)

J S = J S√

µ/ε

2(42)

where J S is the straight-ray Jacobian matrix shown in eq. (3). Underthis transformation, J S, J F and J FW have the same units.

The objective of the inversion is to find an optimal set of conduc-tivity perturbations (δσ c) that minimizes in the least-squares sensethe difference between the data (δD) and the model output (Jδσ c).To examine the effect of scattering theory and straight-ray physicson the inverse solution we avoid incorporating a priori informationinto the solution, and therefore only data misfit contributes to ourobjective function. The objective function can be written as

min ||Jδσc − δD||2. (43)

The standard least-squares solution is obtained by solving the nor-mal equations given by

J T Jδσ = J TδD. (44)

Using SVD, J can be decomposed into (Golub & Van Loan 1983)

J = U�V T (45)

where U is a matrix of singular vectors in the data space, V is amatrix of singular vectors (or basis functions) in the model spaceand � is an ordered diagonal matrix of decreasing singular values.For the model geometry in our synthetic example the reconstructedsolution δσ c is given by

δσc = V V Tδσ = V �−1UrmT δD (46)

where VV T is the resolution matrix and δσ is the exact solution.Eq. (46) can be written alternatively as

δσc =k∑

j=1

(U T

j δD

� j

)Vi (47)

C© 2005 RAS, GJI, 162, 9–24


where k is the truncation index. For example, if λk+1 ≈ 0 then kwould be the maximum truncation index for the truncated SVD so-lution and the resolution matrix will not be identity (i.e. VV T �= I ).Eq. (47) shows that the reconstructed solution is a linear combina-tion of basis functions Vi weighted by the coefficients U T

j δD/� j .To compare the scattering theory and ray-based Jacobian matriceswith the full-waveform Jacobian matrix, we show the model ba-sis functions and singular values as a function of truncation index.The full-waveform Jacobian matrix represents the true physics aswell as possible and is the benchmark with which we compare thescattering theory and ray-based Jacobian matrices. The rigour ofeither method is evaluated based on how well the basis functionand singular value properties match those of their full-waveformcounterparts. We also show Fresnel zone and ray-based δσ c distri-

JFij = 2.07

Time (ns)Time (ns)

BKGNDTOTAL

BKGNDTOTAL

F.D Sol.

BKGNDTOTAL

BKGNDTOTAL

Am

plitu

deA

mpl

itude

Am

plitu

de

a

b c

d e

f g

S.T. Sol.F.D Sol.

JFij = 2.10

S.T. Sol.

mS m−1σ=2

Distance (m)

Transmitter Receiver

Dep

th (

m)

(25 cm x 25 cm)δσ= 20 mS m−1

velocity = 0.099 m ns−1

(1st Fresnel zone boundary)= 74.4 ns iso–time contour

−0.07−0.05−0.03−0.010.010.030.050.07

−0.75

−0.45

−0.15

0.15

60 70 80 90 100 60 70 80 90 100

64 66 68 70 72 74 76 64 66 68 70 72 74 76−0.01

0.03

0.07

0.11

0.15

tsr tfp

tfp

tsr

−0.80

−0.20

0.40

1.00

+ +

876543210−14

3

2

1

0

tfp

Figure 4. Scattering theory (S.T.) and finite-difference (F.D.) solution comparisons. (a) Source position, receiver position, scattering size, magnitude andFresnel volume boundary. (b) Scattering theory and finite-difference background solutions ei (σ a (r)). (c) Scattering theory and finite-difference scatteredsolutions e1i (σ a(r) + δσ j ). (d) Scattering theory background and total field solutions. (e) Finite-difference background and total field solutions. (f) Scatteringtheory background and total field first pulse, integration limits and sensitivity estimation. (g) Finite-difference background and total field first pulse, integrationlimits and sensitivity estimation. The sensitivity estimations are comparable in each case.

butions at several truncation points to illustrate how the solution isbeing constructed as more basis functions are included in the so-lution. This analysis yields insight into how and why the physicalimprovement represented by the Fresnel zone inversion providesa more resolved solution with fewer artefacts than the ray-basedinversion.

4 R E S U LT S A N D D I S C U S S I O N

4.1 Sensitivity distributions, radiation pattern effects,and data predictions

Fig. 4 shows the scattering theory and finite-difference solutions forei(σ (r)), ei(σ (r) + δσ j ) and e1i (σ (r) + δσ j ). The finite-difference

C© 2005 RAS, GJI, 162, 9–24


grid consists of 5 cm2 cells which are small enough to effectivelyremove numerical dispersion (Holliger & Bergmann 2002). A largeδσ j was chosen to illustrate the difference in first-pulse amplitudebetween ei(σ (r)) and ei(σ (r) + δσ j ) and to show the close proxim-ity of the scattering theory solution to the finite-difference solutionwhen δσ j is large. Although the conductivity difference is an orderof magnitude greater than the background conductivity, the rela-tive background and scattered field amplitudes are approximatelyequal for each solution. This result suggests that for the case of asingle scatterer the second- and higher-order terms of eq. (9) are in-significant, and that most of the sensitivity errors are due to multiplescattering effects that become operative in heterogeneous conductiv-ity distributions. Although the Fourier amplitudes of each solutionare identical, there is a slight difference in the phase of the solu-tions which is due to the manner in which the source functions areimplemented. These phase differences are of little consequence inthe computation of J F because the phase of each background andscattered solution depends on the phase of the source function suchthat the effect on the first-pulse sensitivity of the total solution is in-dependent of phase. Thus, the sensitivity of the change in first-pulseenergy to the scatterer is comparable between the scattering theoryand finite-difference solutions.

Fig. 5 shows the 2-D distribution of Fresnel zone sensitivitiesand compares the scattering theory and finite-difference sensitivitydistributions about a homogeneous background. The distributionsare similar, with peak sensitivities near the source and receiver posi-tions. Sensitivities are depressed along the ray path where ray theoryconcentrates all of the sensitivity. The largest errors in the sensitivityapproximations occur near the source and receiver positions whereequipotential surface curvatures are large and the assumption of aconstant electric field within cells may be violated. Reducing thesize of the cells near the antenna positions will reduce these er-rors. Larger errors may occur when the model is heterogeneous andmultiple scattering becomes operative. In practical applications thetrue conductivity distribution will always be heterogeneous (or elsethere would be nothing to image) so that multiple scattering er-rors arise in the scattering theory sensitivities. We expect the finite-difference sensitivities to be more accurate than the scattering theorysensitivities in this case. However, the scattering theory sensitivitiesare more efficient to compute. In Fig. 5 the finite-difference sensi-tivity distribution takes approximately 90 min to compute whereasthe scattering distribution takes approximately 1 s on the same com-puter. As a result, we parallelized the finite-difference forward codeto expedite the computation of J FW. We will show in the SVD anal-ysis that the scattering theory sensitivities in J F have properties (i.e.singular values and basis functions) similar to the exact sensitivitiesin J FW computed with the finite-difference solution. This suggeststhat under the appropriate circumstances J F can provide inverse es-timates similar to those provided by J FW with a dramatic increasein computational efficiency.

Scattering theory and finite-difference sensitivity distributionsfor a high-angle source–receiver configuration and dipole-type ra-diation pattern are shown in Fig. 6. The dipole radiation patterncauses the sensitivity distribution to skew toward cells that are ori-ented horizontally with respect to the source and receiver. Fig. 7shows isosurface and cross-sectional representations of 3-D scatter-ing theory Fresnel volume sensitivity distributions for isotropic anddipole-type radiation patterns. In the isotropic case the sensitivitydistribution is symmetric about the corresponding straight-ray andinsensitive along the ray path, assuming a doughnut-type shape inplanes normal to the ray path. Fig. 7(b) show the sensitivity distri-bution when dipole type radiation patterns are implemented. The

Figure 5. Scattering theory and finite-difference Fresnel zone sensitivitydistribution comparisons for 100 MHz source, background conductivity of0.002 S m−1 and a scattering strength of δσ = 0.0001 S m−1 . Cell sizes are0.25 cm for the scattering theory solution and 0.05 cm for the finite-differencesolution. (a) Finite-difference sensitivity distribution for a homogeneousbackground. (b) Scattering theory sensitivity distribution. (c) Sensitivityerror

distribution normal to the corresponding ray path assumes ahorseshoe-type shape with the maximum sensitivities skewed to-wards horizontal.

As discussed in Section 2.1, differencing attenuation data causesradiation pattern effects to cancel when waves propagate as rays andradiation patterns are time invariant. This is only true in infinite-frequency propagation. Figs 6 and 7 show that finite-frequencyattenuation differences are influenced by radiation pattern effectsbecause the sensitivities of attenuation differences within theFresnel volume are influenced by radiation patterns. Assuming thatradiation patterns cancel through data differencing could cause sig-nificant forward model error and inverse solution artefacts in thestraight-ray case. Scattering theory sensitivities can account forthese effects if the radiation patterns are known or can be adequatelyapproximated.

Another interesting aspect of the sensitivity distributions are thedepressed sensitivities along the ray path in the 2-D case and the

C© 2005 RAS, GJI, 162, 9–24


Figure 6. High-angle Fresnel zone sensitivities showing the effect of a dipole-type radiation pattern. (a) Finite-difference sensitivity distribution. (b) Scatteringtheory sensitivity distribution.

Figure 7. 3-D scattering theory Fresnel volumes. Isosurfaces are at a sensitivity value of 1. Source (x , y, z) = (1, 0, −1) m. Receiver (x , y, z) = (7, 0, −15).The cross-section is z-normal at z = −12 m. (a) Sensitivities assuming an isotropic radiation pattern. (b) Sensitivities assuming a dipole-type source and receiverradiation patterns.

complete insensitivity along the ray path in the 3-D case. Similarvelocity sensitivity distributions have been documented in seismicwave theory (Marquering et al. 1999). However, the reason for theinsensitivity along the ray is different in this case. In the seismiccase, energy scattered from points along the ray arrives in phasewith the background wavelet so that only amplitudes are affectedin the total wavelet (Hung et al. 2001). Thus there is no traveltimeshift required to correlate the background and total wavelets and nosensitivity along the ray. In the EM wave propagation case, the scat-tered wavelet from a conductive scatterer arrives 90◦ out of phasewith the background wavelet for points along the ray. This is illus-trated by the equation for the total field given in eq. (35). For pointsalong the ray the complex exponential terms in the background andscattered wavelets are equal. However, the background wavelet coef-

ficient contains the complex number i and the scattered wavelet doesnot. Thus the scattered wavelet is phase shifted with respect to thebackground wavelet for points along the ray. This phase shift causesequal amounts of constructive and destructive energy to be addedto the first pulse of the background wavelet, which is the cause ofthe insensitivity of conductive scatterers along the ray. The effect isdisplayed graphically in Fig. 8 which shows a background and scat-tered wavelet for a scatterer placed on the ray. The zero-sensitivityregion can be distorted when radiation patterns are anisotropic, asshown in Fig. 7.To compare the accuracy of the Fresnel zone and ray-based for-ward models, we used the finite-difference solution to generate aset of background traces for a homogeneous medium. These tracescorrespond to ea,i in eq. (5). Next we added conductive anomalies

C© 2005 RAS, GJI, 162, 9–24


t fp

Background

Scattered

0

0

First Pulse

Not to scale

Not to scale

Am

plitu

de

Am

plitu

de

Time (s)98e–9 106e–9 114e–990e–9

Figure 8. Example background and scattered wavelet for a scatterer lo-cated along the ray path. The scattered wavelet is phase shifted and addsapproximately equal amounts of constructive and destructive energy to thefirst pulse of the background wavelet, resulting in depressed sensitivity alongthe ray.

representing a tracer or contaminant plume (i.e. δσ (r)) to the back-ground and generated the traces representing eb,i . The ‘true’ syn-thetic data δDi were then computed by eq. (5). To compute theFresnel zone and straight-ray predictions of δDi, (δDF

i and δDSi re-

spectively), we computed J F and J S and then the predicted databy

δDFi =

∑j

J Fi jδσ j and δDS

i =∑

j

J Si jδσ j . (48)

Fig. 9 shows the δσ distribution, the true data cross-plot, and theFresnel zone and ray-based data prediction and prediction errorcross-plots.

The data prediction and prediction error cross-plots in Fig. 9are useful for evaluating the accuracy of each model. If the modelperforms well, both the shape and magnitude of the predicted datacross-plot will match that of the true data cross-plot. The Fresnelzone model tends to over-predict some of the data, but in generalthe shape of the distribution matches that of the true data reasonablywell. In comparison, the ray-based model significantly mis-predictsboth the magnitude and the distribution of the data. Thus, the Fresnelzone inverse solution will fit the data well when the inverse solutionδσ c is approximately equal to the true solution δσ . This is not thecase for the straight-ray model, and the straight-ray inversion mayimpose artefacts in order to fit the data.

This point is further illustrated by a thought experiment. Fig. 10compares J F and J S versus cell number for a particular shot–receiver pair. The total integrated sensitivity for each case is ap-proximately equal. This is a requirement since, under low-loss con-

0 2 4 6 8 10 12 14

0

2

4

6

8

10

12

14

2

0

4

68

10

12

14

0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 14

2

0

4

68

10

12

14

10 2 3 4 5Log Energy Difference

0 2 4 6 8 10 12 14

2

0

4

68

10

12

14

0 2 4 6 8 10 12 14

1 0 −1 −2 −3Prediction Error

e

Rec

eive

r D

epth

(m

)

f

Source Depth (m) Source Depth (m)

c

a

Source Depth (m)

Rec

eive

r D

epth

(m

)

Rec

eive

r D

epth

(m

)

d

b

Source Depth (m)

Distance (m) Source Depth (m)

Dep

th (

m)

0 2 4 6

rece

iver

s

sour

ces

.01

0

.01

.02

.03

∆C

ondu

ctiv

ity (

S m

−1)

Figure 9. Fresnel zone and ray-based data predictions. The backgroundconductivity is 0.002 S m−1. (a) The conductivity model δσ (r). Boreholeseparation is 7 m. Boreholes are 15 m deep with sources and receivers every0.25 m. (b) Finite-difference-generated data (δDi) cross-plot. (c) Scatteringtheory data prediction (δDF

i ) cross-plot. (d) Ray-based data prediction (δDSi )

cross-plot. (e) Scattering theory prediction error (δDFi − δDi). (f) Ray-based

model prediction error (δDSi − δDi).

ditions, the Fresnel zone and straight-ray models must predict thesame data for large-scale δσ (r) distributions, such as when δσ (r)is homogeneous (Spetzler & Snieder 2004). However, because theray-based sensitivities are concentrated along the ray, cells throughwhich the ray passes have erroneously high sensitivities, causingthe large data over-predictions shown in Fig. 9(d). Cells within theFresnel zone but not directly in the path of the ray have zero sensi-tivity in the ray-based case, which is erroneously low, causing thedata underpredictions shown in Fig. 9(d). Now consider the caseof the two shot–receiver pairs shown in Fig. 10(b). In the ray 1case, the ray does not directly pass through the conductive anomaly.

C© 2005 RAS, GJI, 162, 9–24


Straight RayFresnel Zone

Sen

sitiv

ity

Cell Number

SourceWell

ReceiverWell

Conductive Anomaly

a

bRay / Fres. Zone 1

Ray / Fres. Zone 2

0

2

4

6

8

10

12

14

16

18

20

22

24

900 1000 1100 1200 1300 1400 1500 1600 1700

Figure 10. (a) Fresnel zone versus straight-ray sensitivity magnitudes ver-sus parameter number for a particular source–receiver pair. The total sensitiv-ity for each case is equal. Straight-ray sensitivities are too large and spiked atcells through which the ray passes. (b) Schematic of source–receiver config-urations, the associated straight-ray and Fresnel zone sensitivity boundaries,and a conductive anomaly.

However, the anomaly is within the Fresnel zone so the datumassociated with that ray will be affected by the anomaly. In orderto fit the datum associated with that ray, the ray-based inversionmust smear the boundary of the anomaly to the ray which resultsin a loss of spatial resolution. The ray-based inversion is forced toexpand the boundaries of the estimated anomaly to fit the data asso-ciated with rays passing near the true boundaries. This smearing is

Straight Ray

Mag

nitu

de

Singular Value

Fresnel Zone

Full Waveform

2

××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

×××××××

×

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

0 20 40 60 80 100 120 140 160 180 200 220 240 26010

10

10

10

10

10

10

−1

0

1

3

4

5

+++

++

×++×

Figure 11. Singular value comparison for full-waveform (J FWi j ), scattering theory (J F

i j ) and ray-based (J Si j ) decomposition.

a consequence of model error and increases as frequency decreases,Fresnel zones become wider, and the ray-geometric approximationbecomes less valid.

Next consider ray 2 in Fig. 10(b). In this case, the ray passesdirectly through the anomaly and the anomaly encompasses only apart of the Fresnel zone. The ray-based inversion is not forced toexpand the boundaries of the estimated plume to fit the datum inthis case, but because the sensitivities along the ray are too high, thecells through which this ray passes must have values of conductiv-ity change that are too low or oscillatory in order to fit the datumassociated with the ray. The overall result in the inverse estimate isthat the estimated values of conductivity change are too low. TheSVD solutions we will show illustrate both the boundary smearingeffect and the low predictions caused by model error in the ray-basedinversions.

4.2 SVD analysis

To illustrate how physical improvement in the Fresnel zone repre-sentation of wave propagation leads to more accurate and resolvedattenuation-difference estimates, we conducted a SVD analysis ofa synthetic case. We constructed J FW, J F and J S for the model ge-ometry shown in Fig. 9(a), only with sources and receivers at 1 mintervals from 0 to 15 m for a total of 256 data points. Recall J FW,J F and J S are respectively the full waveform, scattering theory andray-based Jacobian matrices. J FW was computed about the true so-lution and contains the exact sensitivities. Parameter cells are 0.25 msquares for a total of 1680 parameters. Fig. 11 shows the non-zerosingular values corresponding to J FW, J F and J S as a function ofincreasing index number. The spectrum shows that the singular val-ues begin at approximately the same magnitude of 8 × 104. Thestraight-ray singular values initially drop and then cluster aroundthe value of approximately 104 whereas the full-waveform and scat-tering theory singular values steadily decrease, displaying a decay inthe singular value spectrum. Based on the singular values alone, wemight conclude that the ray-based method will provide a better solu-tion, since the singular values for the ray-based case are greater thanthe singular values for the scattering theory case. This would be trueif the nature of the basis functions Vi were comparable between thetwo methods. However, these basis functions show a significantlydifferent character. Fig. 12 shows the basis functions correspondingto selected singular values in Fig. 11. The full-waveform and scat-tering theory basis functions are similar, beginning with a smoothslowly varying nature and gradually become more oscillatory withincreasing index. In addition, these basis functions display more lo-calization than the straight-ray counterparts. For example, the firstbasis function for the straight-ray case is essentially providing a

C© 2005 RAS, GJI, 162, 9–24


Figure 12. Full-waveform, Fresnel zone and straight-ray basis function (Vi) comparisons from SVD analysis. Columns A show the full-waveform basisfunctions computed about the solution. Columns B and C show the corresponding scattering theory and straight-ray basis functions respectively. The indicescorrespond to the singular values in Fig. 11 and the colour scale is constant for a given index.

measure of ray coverage, where the corresponding full-waveformand scattering theory basis functions are more localized and exhibitless X-pattern smearing typical of ray-based methods. With respectto the first J F basis function, the first J FW basis function is distorteddisplaying the effects of the heterogeneous background that are notaccounted for in the scattering theory approximation. However, thefull-waveform and scattering theory basis functions display a simi-lar structure at each index. The straight-ray basis functions quickly

become oscillatory and less localized and are dominated by theX-pattern feature. As the basis functions are scaled and added toconstruct the inverse estimates, the X-patterns transfer to the solu-tion as artefacts. This suggests that the X-pattern artefacts persistentin many ray-based geophysical tomographic inversions are not onlycaused by limited aperture data collection geometries, but are exac-erbated by the ray approximation. Based on the nature of the basisfunctions shown in Fig. 12 we would expect fewer of these artefacts

C© 2005 RAS, GJI, 162, 9–24


in the scattering theory solution even though the data collectiongeometry is the same in both cases. This X-pattern is also evident inhigher-order scattering theory basis functions, but is more prevalentin the straight-ray case.

We used the finite-difference solution to compute the data for thiscase and added normally distributed noise with a standard deviation(S.D.) of 5 per cent of the maximum data value. To evaluate howwell the predicted data fit the true data we use the chi-squared (χ2)criterion

χ 2 = ‖Wd(δDtrue − δDpred)‖2 (49)

where W dis a diagonal matrix of 1/S.D., δD trueis the true data andδDpredis the predicted data. When the data are appropriately fittedthe expected value of χ 2is equal to the number of data points, whichis 256 in this case. Fig. 13 shows the χ 2value versus truncation indexfor the truncated scattering theory and straight-ray SVD solutions.The χ2prediction errors decrease more quickly with truncation in-dex in the scattering theory inversion, indicating that the Fresnelzone inversion is able to fit the data with fewer high-order oscilla-tory basis functions, limiting the build-up of artefacts. This is alsoevident in the singular value spectrum. The clustering of singularvalues for the straight-ray case would indicate that if the singu-lar value magnitude corresponding to the noise level is below thisclustering then high-index oscillatory basis functions would con-tribute to the solution. However, the decay of the singular valuesin the scattering theory case indicates that given the same singularvalue magnitude, the Fresnel zone solution would introduce fewerhigh-index oscillatory basis functions in the solution, reducing thebuild-up of artefacts. This is intuitively appealing since the physicsof EM wave propagation are better represented by the Fresnel zonesensitivity distribution. Thus a smaller number of good basis func-tions are able to fit the data better than a larger number of poor basisfunctions.

Fig. 14 shows the construction of the SVD solutions for eachmethod at several truncation indices. The magnitude of the con-ductivity difference values is immediately evident. The straight-raysolutions underpredict true conductivity difference values. As dis-cussed previously, these estimates are too low because the ray pathsensitivities are too large. Low ray-based inverse estimates were alsonoted by Holliger & Bergmann (2002) who used synthetic data andray-based inversions to estimate attenuation coefficient values for aknown model. The scattering theory solutions also underpredict thetrue values, but to a lesser degree. With respect to spatial resolutionthe scattering solution is able to distinguish between the two centreanomalies while the straight-ray solution is unable to localize the

Chi

–squ

ared

Val

ue

Chi–squared value vs. Truncation

Fresnel Zone

Straight Ray

Target = 256

156 240

Truncation Index

0 40 80 120 160 200 240 2800

10

1

10

2

10

3

10

4

10

5

10

6

10

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

×

Figure 13. Comparison of Fresnel zone and straight-ray χ2 values versus truncation index. Normally distributed noise was added to data with a standarddeviation of 5 per cent of the maximum data value. The data are appropriately fitted when χ2 reaches a target value of 256.

two peaks. The scattering solutions also resolve the upper anomalymore effectively, while the straight-ray solution tends to smear theboundaries and obscure the peak. The boundaries of the straight-rayestimates are expanded with respect to the true solution becauseray theory does not account for the sensitivities of cells adjacent tothe rays. This is particularly evident for smaller-scale features suchas the anomaly at approximately 5 m depth in Fig. 9(a). Althoughsomewhat hidden by the colour scale, the straight-ray solutions aremore significantly marked by X-pattern artefacts as expected. Asthe truncation indices increase toward the target index for eachmethod, the scattering theory solutions become more resolved whilethe straight-ray solutions change slowly and are unable to resolvethe three separate peaks.

We have shown that under conditions common to georadar tomo-graphic surveys, the physics of wave propagation represented by J F

are more accurate than the physics represented by J S. This physicalimprovement results in more resolved tomograms, both in terms ofthe magnitude and spatial distribution of bulk conductivity changes.But how well do inverse tomographic estimates using J F comparewith the best possible estimates we might obtain by using a Jaco-bian matrix with insignificant sensitivity errors such as J FW? Wecould answer this question for our synthetic example by conductingthe full-waveform inversion using the finite-difference solution tocompute J FW. The full-waveform solution would probably requireseveral iterations, each with an update of J FW. We computed J FW

on the Beowulf cluster at Boise State University. The computationof J FW for a single iteration required 1.5 days of parallel computa-tion time on 100 2.4 GHz Pentium IV processors. Thus we deemedthe full inversion an excessive computational burden. In contrast,computation of J F for our synthetic problem only required approx-imately 10 min on an single processor machine running a 2.2 GHzPentium IV processor. This represents immense computational sav-ings at the expense of losing some accuracy in the sensitivity approx-imations. Although the full-waveform solution was not computed,comparison of the basis functions and singular values for J F andJ FW gives insight into the accuracy lost by approximating J FW withJ F. Although the effects of heterogeneity in δσ (r) are evident inthe full-waveform basis functions, the full-waveform and scatteringtheory basis functions display similar properties in terms of shapeand magnitude. In addition, the singular value spectra for each caseare nearly identical. When the inverse estimates are constructed, wehave similar basis functions being scaled by similar singular valuesand added to construct the solutions. Thus, the inverse estimatesprovided by J F will probably be similar to those provided by J FW

with a significant decrease in computational effort.

C© 2005 RAS, GJI, 162, 9–24


Figure 14. Fresnel zone and straight-ray SVD solutions versus truncation index.

5 C O N C L U S I O N

We have presented an efficient method of approximating Fresnelvolume sensitivity distributions for use in georadar attenuation-difference tomography in environments where dielectric contrasts

are small and ray representations of wave propagation are straight.The characteristics of these distributions are similar to those shownby Marquering et al. (1999) in seismic propagation problems, butfor a different reason. Namely, the distributions display the paradox-ical result that the Fresnel volume sensitivities are zero along the ray

C© 2005 RAS, GJI, 162, 9–24


path. We have included in the approximation the effects of radiationpatterns and have shown that radiation patterns affect attenuation-difference data and are important in attenuation-difference tomog-raphy. The SVD analysis provided several interesting insights intocommon observations in ray-based inverse estimates. In particular,ray-based basis functions tend to impose the X-pattern artefact onthe inverse solution. We have also shown with a data predictionexample that the Fresnel zone forward model represents EM wavepropagation better than the ray-based model. In the SVD analysis, weshowed how this physical improvement in the forward model resultsin more accurate and better resolved inverse estimates of bulk con-ductivity changes. The physical improvement is most evident whencomparing the scattering theory and the full-waveform model basisfunctions and singular values which display similar behaviour. Whendielectric heterogeneity is insignificant, bulk conductivity changesare relatively small, and EM waves propagate under low-loss con-ditions, Fresnel volume sensitivities approximated with scatteringtheory can provide inverse estimates that are good approximationsto the solutions provided by full-waveform inversions with a signif-icant decrease in computational effort.

A C K N O W L E D G M E N T S

We gratefully acknowledge the constructive review provided byGuust Nolet, who helped us clarify our discussion of the sensitivitydistributions. A second anonymous reviewer also provided usefulinsights and comments and helped us correct a misconception con-cerning the Hankel function. Many thanks to Klaus Holliger forgraciously allowing us to employ his finite-difference code in thisproject. Bill Clement, Tom Clemo and John Bradford also providedhelpful comments. Funding for this work was provided by the In-land Northwest Research Alliance (PhD fellowship for TJ), BechtelBWXT Idaho, LLC (contract 00036570), and EPA grant X-970085-01-0. Computations were performed on the geophysical computingsystem (NSF-EPSCoR grant no EPS0132626) at the Center for Geo-physical Investigation of the Shallow Subsurface and the Beowulfcluster at Boise State University (‘Development of Tools to Enablethe Port of Software to a Beowulf Cluster’, NSF-Major Research In-frastructure Award no 0321233). We would also like to thank JamesNelson for his assistance with the parallel computers.

R E F E R E N C E S

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A P P E N D I X A :

The homogeneous Helmholtz equation for the electric field is givenby

∇2E(r, ω) + k2E(r, ω) = 0 (A1)

where k2 = ω2 µ0 ε − iωµ0 σ is the wavenumber (or propagationconstant). If σ = 0 then all currents are displacement currents andthere is no energy lost through Ohmic dissipation. If σ �= 0 then theratio of displacement current flow to Ohmic current flow is describedby the loss tangent which is given by

tan δ = σ

ωε. (A2)

The transition between EM diffusion and wave propagation occursat approximately tan δ = 1. Wave propagation in the georadar regimeis typically low-loss propagation defined by tan δ � 1.

The attenuation coefficient for a sinusoidally varying plane wavein the time domain is given by (Ward & Hohmann 1988)

α = ω√

µε

[1

2(√

1 + (tan δ)2 − 1)

]1/2

. (A3)

Under low-loss conditions, eq. (A3) may be approximated as(Jackson 1999, pp. 295–340)

α ≈ σ

2

√µ

ε(A4)

which indicates that the attenuation coefficient is independentof frequency in the low-loss regime. The relationships between

σ =σ =σ =σ =σ =σ =σ =σ =σ =σ =

0.0500.0450.0400.0350.0300.0250.0200.0150.0010.005

S m−1

Low loss estimate

ε 0

µ0

Atte

nuat

ion

Coe

ff. (

1/m

)

Frequency (MHz)

ε =µ =

10

Transition frequency

0 1 2 310101010

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0

0

××××××××××

++

++

++

++++×

+

Figure A1. Relationship between attenuation coefficient and frequency inthe georadar regime. The attenuation coefficient is frequency independentin the low-loss regime. Wave equation parameters chosen for this case aretypical of those that might be found in a sand and gravel aquifer saturatedwith low-conductivity water.

eq. (A2) through (A4) for a particular case are illustrated in Fig. A1.The curves represent eq. (A3) for several different conductivity val-ues. The ‘+’ symbols show the transition frequencies where tan δ

= 1 for each curve. The ‘×’ symbols show the approximation givenin eq. (A4) valid for the low-loss condition tan δ � 1.

C© 2005 RAS, GJI, 162, 9–24

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