CWP-622

3-D controlled source electromagnetic interferometryby multidimensional deconvolution

Yuanzhong Fan & Roel SniederDepartment of Geophysics and Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA

ABSTRACTControlled Source Electromagnetic (CSEM) is an important technique in hy-drocarbon exploration, because it uses the large contrast in electrical resistivityto distinguishes between water and hydrocarbons. In a shallow sea environment,the airwave that is refracted from the air-water interface dominates the recordedsignal at large offsets. Therefore, the hydrocarbon detection ability of the CSEMis weakened, because the airwave is independent on the properties of the sub-surface. For a layered earth model, we apply multi-dimensional-deconvolutioninterferometry to synthetic 3D CSEM data and estimated the reflection responseof the subsurface. The difference in the models with and without a resistive layeris significantly increased by the employed interferometric analysis. However, therequired receiver spacing is much denser than that of current CSEM surveys.In order to apply this technique in a field survey, we are currently working onhow to relax the required receiver criterion for this technique.

Key words: 3-D CSEM, interferometry, airwave

1 THEORY AND MOTIVATION

1.1 Basic theory and history of virtual sourcetechnique

The concept of interferometry was first introduced tothe seismic community by Jon Claerbout in 1968. Itbecame a hot research topic in geophysics in the lastdecade. The method is also referred to as the virtualsource technique and in a wider sense Green’s functionreconstruction. In this work, we refer to the same tech-nique using these three terms. The key idea of this tech-nique is the following. The Green’s function that de-scribes wave propagation between two receivers can bereconstructed by cross-correlation of the wavefields attwo receiver positions provided that the receivers areenclosed by uncorrelated sources on a closed surface.Because of the advantages of this technique and its usein the passive survey, research on seismic interferom-etry has progressed significantly during the last eightyears Lobkis and Weaver (2001); Weaver and Lobkis(2001); Derode et al. (2003); Campillo and Paul (2003);Weaver and Lobkis (2004); Wapenaar (2004); Snieder(2004); Malcolm et al. (2004); Bakulin and Calvert(2004); Calvert et al. (2004); Wapenaar et al. (2005);

Shapiro et al. (2005); Roux et al. (2005); Sabra et al.(2005a,b); van Wijk (2006); Larose et al. (2006); Bakulinand Calvert (2006); Snieder (2007); Mehta and Snieder(2008).

Snieder (2006) showed that interferometry can beapplied not only to wavefields, but also to diffusivefields. This discovery inspired further research and novelapplications to the diffusive fields, as has happenedfor wave fields. The diffusive fields have a wide rangeof applications and use in physics, chemistry, medi-cal physics, earth science A.Mandelis (1984); Yodh andChance (1995); Basser et al. (1994); Mori and Barkar(1999); Koyama et al. (2006); Constable and Srnka(2007). In earth science, diffusive fields are ubiquitous.Examples include heat conduction, flow in porous me-dia, and low-frequency electromagnetic fields in the con-ductive subsurface. In this work, we focus on the appli-cation of interferometry to low-frequency electromag-netic fields in the subsurface. However, we may easilyextend the concept to other diffusive fields. Because theelectromagnetic field is sensitive to the electric resistiv-ity, it has been used in medical physics and the miningindustry for a long time. In recent years, electromagneticsurvey became increasingly popular in the petroleum in-

42 Y. Fan & R. Snieder

dustry for hydrocarbon exploration because of its abil-ity to distinguish the difference between hydrocarbonand water Andreis and MacGregor (2008); Lien andMannseth (2008). The technique is usually referred toas the Control Source Electromagnetic (CSEM). Seismicmethods are not sensitive to the chemical content porefluids. By using a combination of seismic and CSEMmethods, the success of finding the hydrocarbon reser-voir in the subsurface dramatically increases Darnetet al. (2007); Kwon and Snieder (2009).

The theories of interferometry based on cross-correlation for waves and diffusion are similar but differin the required source distribution. Because of this dif-ference, there is almost no real application of diffusioninterferometry yet.

Interferometry for acoustic waves can be expressedas Snieder et al. (2007)

G(rA, rB , ω) − G∗(rA, rB , ω) =

2iω

IS

1

ρcG(rA, r, ω)G∗(rB , r, ω)dS, (1)

in which G(rA, rB , ω) is the pressure Green’s functioncaused by a volume injection that describes wave prop-agation from rB to rA respectively, ∗ indicates complexconjugation, S is the surface where sources are located,and ω is the angular frequency. For the case of diffusioninterferometry the expression becomes

G(rA, rB , ω) − G∗(rA, rB , ω) =

2iω

ZV

G(rA, r, ω)G∗(rB , r, ω)dV, (2)

where V is the volume containing the sources. The maindifference is that the surface integral in equation (1) be-comes a volume integral in equation (2) Snieder (2006).This volume integral implies that sources in the entirevolume are required to reconstruct the Green’s func-tion. From previous work Fan and Snieder (2008) welearned that the sources in a small volume close to thetwo receivers give the largest contribution to the Green’sfunction reconstruction. In practice, a finite number ofsources is sufficient to reconstruct the Green’s function.Criteria of the required source distribution are derivedin this work. However, it is still unpractical to applycross-correlation-based interferometry to a real applica-tion, because sources close to the receivers are criticalin this technique. The purpose of interferometry tech-nique is to use a receiver both as a receiver and a virtualsource. If real sources close to the receivers are requiredto do this, then we could just as well place a real sourceat the receiver position.

When the receivers are located in a planeand sources are placed above this plane, a multi-dimensional-deconvolution approach is shown to be ap-plicable to a diffusion field Amundsen et al. (2006); Slobet al. (2007); Wapenaar et al. (2008). This approachworks for both diffusion and waves. It also holds forany field which can be decomposed into upgoing and

Figure 1. A generic configuration for the application of themulti-dimensional deconvolution interferometry concept.

downgoing components. A generic geometry for this ap-proach is sketched in Figure 1. The source is denotedby the star, and receivers are located on the plane B1.B0 is a boundary above the sources, which may or maynot be present. D and U represent downgoing and up-going fields, respectively. If we decompose the field intoupgoing and downgoing components, the upgoing anddowngoing fields can be related with the following inte-gral equation

U(xA, xs, ω) =

ZR(xA, x, ω)D(x, xs, ω)dx, (3)

where U(xA, xs, ω) represents the upgoing field receivedat location xA in the frequency domain due to thesource at xs. The downgoing field is noted by D, andR(xA, x, ω) is the reflection response that relates thedowngoing field at x to the upgoing field at xA. Al-though the entire medium can be arbitrarily heteroge-neous in order to decompose the field into upgoing anddowngoing components Grimbergen et al. (1998), theup-down decomposition operator can be simplified, ifthe layer where receivers are located is homogeneous orweakly heterogeneous Wapenaar et al. (2008). Becausethe downgoing fields at all positions in the plane B1

contribute to the upgoing field at position xA, we needto integrate x over the whole surface to obtain a com-plete upgoing field at xA. The inversion of the R fromequation (3) is ill-posed because no unique R can beobtained from a downgoing field D and upgoing field Uexcited by a single source. If a source at another posi-tion xs′ is used, a different pair of U and D is obtainedfrom the decomposition. The medium response R, how-ever, remains the same because it is independent on thesource position. This means that the more sources weuse, the more constraints there are on the inversion ofR. Therefore, a band-limited medium response R can beaccurately inverted from a band-limited input signal, ifa sufficient number of real sources are used.

A discretized version of equation (3) may help toillustrate the importance of multiple sources. Equation

3-D CSEM interferometry 43

(4) is a discretized version of equation (3) for a siglesource at xs0BBB@

U1

U2

...Un

1CCCA =

0BB@R11 R12 · · · R1n

R21 · · · · · · · · ·· · · · · · · · · · · ·Rn1 · · · · · · Rnn

1CCA0BBB@

D1

D2

...Dn

1CCCA , (4)

where the subscript 1 to n of U and D are the discretizedsampling points of the surface B1, Rij denotes the reflec-tion response between the position i and j. The arraysof U and D are the known data and R is the unknownmatrix. In general, there is no unique solution for ma-trix R because there are more unknowns (n2) than thenumber of equation (n). Using reciprocity, the numberof unknowns can be reduced to n(n + 1)/2. The expres-sion below is a similar equation but combines sources atposition xs and another position xs′0BBB@

U1, U ′1

U2, U ′2

...Un, U ′

n

1CCCA =

0BB@R11 R12 · · · R1n

R21 · · · · · · · · ·· · · · · · · · · · · ·Rn1 · · · · · · Rnn

1CCA0BBB@

D1, D′1

D2, D′2

...Dn, D′

n

1CCCA ,(5)

where U ′ and D′ are the new upgoing and downgoingfields generated by the source at position xs′ . The ma-trix R remains the same as it is independent on thesource. By increasing the number of sources, the num-ber of columns in U and D increases. The matrix R canbe accurately estimated if a sufficient number of sourcesare used. However, this calculated maxtrix R does notnecessarily represent the real medium response R.

When discretizing equation (3), only a finite num-ber of the receivers from a limited range of the surfaceB1 can be used. This raises the question how to choosethe receiver distribution in order to represent the inte-gral in equation (3) accurately for a band-limited re-sponse R (receiver density and the range of the surfacewhere the receiver are located)?

1.2 Why do we apply interferometry inmarine CSEM?

Figure 2 shows a typical configuration of an offshoremarine CSEM survey. A resistive layer (e.g hydrocar-bons) in the subsurface, acts as a secondary source thatgenerates an upgoing EM field. We can distinguish be-tween models with and without the resistive layer fromthe secondary fields which the subsurface generates. Thelarge difference in the electrical resistivity between wa-ter and hydrocarbons makes CSEM an accurate tool todistinguish between these pore fluids. Most of the cur-rent successful applications of CSEM are offshore be-cause the water strongly attenuates anthropogenic andnatural noise. However, one of the most significant prob-lems in offshore CSEM is the airwave when the waterlayer is shallow. The airwave is the secondary EM fieldrefracted from the water-air interface as shown in figure

Figure 2. A simple configuration of a offshore CSEM surveyin a layered earth model.

Figure 3. The configuration of CSEM after the applicationof the multi-dimentional-deconvolution interferometry.

2. The airwave weakens the difference between the sig-nal with a target layer and the signal without a targetlayer because it is much stronger than the target signal.

If we can successfully apply the multi-dimension-deconvolution interferometry as described in the lastsection to CSEM, one of the receivers is converted into asource and the overburden is extended upwards to a ho-mogeneous half space Wapenaar et al. (2008). The newconfiguration after applying this technique is shown infigure 3. In this configuration, the air-water interfaceand the sea floor are removed, and there is no secondaryfield is refracted from the medium above the receivers.Note that the sea floor interface may or may not beremoved depending on the boundary condition whichwe use in the decomposition process. Therefore, by ap-

44 Y. Fan & R. Snieder

Figure 4. The layered earth model used in the syntheticexample.

plying this interferometry technique to CSEM data, theairwave problem is solved and the complexity of themedia above the receiver layer is removed as well. Con-sequently, the secondary field is generated only by thesubsurface and therefore it is easier and more accurateto detect the properties of a target layer. This signif-icant improvement has been demonstrated in 2D syn-thetic examples (Wapenaar et al. 2008; Hunziker et al.2009).

2 SYNTHETIC EXAMPLE IN A LAYEREDEARTH MODEL

A synthetic study of the described multi-dimentional-deconvolution interferometry has been applied to a 3Dmarine CSEM survey in a layered earth model. Thissynthetic study shows the feasibility of applying thistechnique to the CSEM data in such an earth modeland illustrate how this technique helps to detect the tar-get layer. This study also provides the required receiverdistribution.

The layered model used is shown in figure 4.Because both wavenumbers in the x and y directions arerequired in the decomposition into upgoing and down-going fields, and the surface integral in the equation (3)must be replaced by summation of receivers, a 2D re-ceiver array is used in the synthetic example as shownin figure 5. The layered model used is shown in figure4. The uniformly sampled 2D receiver array is on thesea floor from the position (-10 km, -10 km) to (-10 km,10 km) with a receiver separation dr of 50 m in both x

Figure 5. A map view of the 2-D receiver array used in thesynthetic example.

Figure 6. Ex field without target layer in the log10 scale.

and y directions. The EM source (arrow in figure 5) is adipole in the x direction with a length of 100 m and a ACcurrent of 100 A and an operating frequency of 0.25 Hz.The source is located 100 m above the sea floor. Theemployed station spacing is unrealistically dense, andthe current work is aimed at increasing the spacing.

The inline electric field, in this case is Ex, is shownin figures 6 and 7. Figure 6 is the Ex field without thetarget layer and figure 7 is the Ex field with the tar-get layer. The difference of the total Ex field betweenthe models with and without the target layer is small.The difference occurs at an angle of about 45 degrees.However, we normally do not use the signal in this direc-tion because of its weakness. In current CSEM surveys,it is common to use only the inline profile of the Ex

field because the signal to noise ratio is high and 2Dacquisition is more expensive. Figure 8 shows the inlineprofile of the Ex field. The solid curve is the signal with-out the target layer and dashed curve is the signal withthe target layer. For small offsets (< 2 km), there is

3-D CSEM interferometry 45

Figure 7. Ex field with target layer in the log10 scale.

−10 −5 0 5 10−10

−9

−8

−7

−6

−5

−4

−3

x (km)

Ex (

sV/m

)

without target

with target

Figure 8. The inline profile of the Ex field in the log10 scale.

almost no difference between these two curves becausethe direct field dominates. For large offsets (> 7 km),the electric field is strongly influnced by the airwave,which does not depend on the subsurface properties atall. Consequently, targets leave useful imprint only forintermediate offsets (2km to 7km). Because this inter-mediate range is narrow and the difference between thesignals with and without target is weak, it is difficultto interpret the difference between the signals with andwithout the target, especially in the presence of noise.

We next, apply the multi-dimentional-deconvolution interferometry to these syntheticdata. The first step of this technique is to decomposethe total field into upgoing and downgoing components.The implementation of the up-down decompositionfollows the theory in the appendix of Wapenaar et al.(2008). Note that the up-down decomposed field isthe square root of energy flux, not the E (electric)or H (magnetic) field. The input data used in the

−10 −5 0 5 10−8

−7

−6

−5

−4

−3

−2

x (km)

inlin

e do

wng

oing

fiel

d ((

sJ) 1

/2/m

)

without targetwith target

Figure 9. The inline profile of the downgoing fields with andwithout the target using the water parameters.

decomposition are the horizontal E and H fields. Themeasured electric and magnetic fields can be relatedwith the upgoing and downgoing flux with the formula

P = L−1 Q (6)

where P is the decomposed upgoing and downgoing po-tential, normalized to energy flux, Q contains the inputhorizontal E and H fields and L−1 is the conversionoperator. Wapenaar et al. (2008) shows a numerical ex-ample for a 2D field with a layered model. With an inlinedipole source, the physical meaning of P = [Pd, Pu]′ isthe decomposed energy flux of the TM (transverse mag-netic) mode (subscripts d and u represent the downgo-ing and upgoing, respectivly). The downgoing field isdefined as the field which decays downwards and theupgoing field is defined as the field which decays up-wards.

In our synthetic example, the field is 3D, henceQ contains four components Ex, Ey, Hx, Hy and P hasfour components as well [P 1

d , P 1u , P 2

d , P 2u ].Because the re-

ceivers are located at the boundary of the water and thesea floor, we can choose the parameters for L−1 fromthe upper medium (water) or the lower medium (seafloor) in the process of the field decomposition. Thesetwo choices of the medium parameters lead to a differ-ent physical meaning for the decomposed field. Usingthe water parameters for the up-down decomposition,we obtain the upgoing and downgoing fields in the wa-ter just above the sea bottom. If the sea floor parametersare used, we obtain the upgoing and downgoing fields inthe sea bottom just below the acquisition surface. Fig-ures 9 to 12 compare the upoing and downgoing fieldsin these two choices. For demonstration purposes, onlythe inline profile of the fields is shown. However, we doneed to calculate the upgoing and downgoing fields forall the receivers in the 2D array. When the water pa-rameters are used in the decomposition, the difference

46 Y. Fan & R. Snieder

−10 −5 0 5 10−9

−8

−7

−6

−5

−4

−3

−2

x (km)

inlin

e up

goin

g fie

ld (

(sJ)

1/2

/m)

without targetwith target

Figure 10. The inline profile of the upgoing fields with andwithout the target using the water parameters.

−10 −5 0 5 10−8

−7

−6

−5

−4

−3

−2

x (km)

inlin

e do

wng

oing

fiel

d ((

sJ)1/

2 /m)

without targetwith target

Figure 11. The inline profile of the downgoing fields withand without the target using the sea floor parameters.

in the downgoing fields with and without the target issmall, as we expected (figure 9). One might question,however, where these slight difference in the downgo-ing field come from. One of the possible answers is themultiples from the air-water interface (cross talk). Theupgoing target signal and the signal refracted from thesea floor are refracted back from the air water interfaceto become a downgoing field. When the water is deeperand less upgoing signal comes back from the water airinterface, this difference becomes smaller. The differencein the upgoing field (figure 10) is clear and the airwavestarts to dominate when the offset becomes large. Thisairwave is refracted from the sea bottom and becomesan upgoing filed.

When the sea floor parameters are used in the de-composition, the slight difference in the downgoing fields

−10 −5 0 5 10−10

−9

−8

−7

−6

−5

−4

x (km)

inlin

e up

goin

g fie

ld (

(sJ)

1/2

/m)

without targetwith target

Figure 12. The inline profile of the upgoing fields with andwithout the target using the sea floor parameters.

(figure 11) might be caused by the refraction at thesea bottom, because when the upgoing target signal en-counters the sea bottom, it is refracted downwards asa downgoing field. The upgoing fields highlight the dif-ference between the models with and without the tar-get because the upgoing field in the sea floor is mainlyfrom the target in the subsurface. In contrast to figure10 where the airwave reflected from the sea bottom isstrong, the airwave leaves a small imprint on the upgo-ing field in figure 12.

After decomposing the energy flux into upgoingand downgoing components, we calculate the impulseresponse of the subsurface using equation (3). This isa multi-dimensional-deconvolution problem to computeR from U and D. For a 1D model, the impulse responseonly depends on the vector which connects two posi-tions. So we can rewrite equation (3) as

U(x, y) =ZR(x − x′, y − y′)D(x′ − xs, y

′ − ys)dx′dy′. (7)

The frequency ω is not shown in the equation. Equa-tion (7) presents a spatial convolution. In wave-numberdomain, we can present spatial convolution in equation(7) by multiplication as

U(kx, ky) = R(kx, ky)D(kx, ky). (8)

Consequently, R can be obtained by a division in thewave number domain. To stabilize this process, the de-vision is modified in the following way.

R(k) =U(k)D∗(k)

D(k)D∗(k) + ε2P

k′ D(k′)D∗(k′)Nk′

, (9)

where ε is a small number and Nk′ is the number ofdiscrete wavenumbers used in the calculation. We useda ε of 10−1.

The up and downgoing field in the sea floor are

3-D CSEM interferometry 47

Figure 13. The impulse reflection response R without targetin the log10 scale.

Figure 14. The impulse reflection response R with target inthe log10 scale.

used in equation (9) to calculate the impulse response.The calculated impulse responses are shown in figure13 (without target) and figure 14 (with target). Thedifference of the impulse response is significant betweenthe two models with and without target. Comparingwith the inline profile of the total Ex field (figure 8), theinline profile of the impulse response (figure 15) givesmuch more pronounced difference between the modelswith and without the target.

3 DISCUSSION

The 3-D synthetic example in this paper shows thatthe virtual source technique in CSEM can significantlyincrease the sensitivity of detecting the high-resistivitylayer (such as hydrocarbon reservoir) in the submarineenvironment. Note that in order to apply this technique

−10 −5 0 5 10

−8

−7

−6

−5

−4

−3

x (km)

refle

ctio

n re

spon

se R

(un

itles

s)

inline reflection response

without targetwith target

Figure 15. The inline profile of the reflection response R inthe log10 scale

−10 −5 0 5 10

−8

−7

−6

−5

−4

−3

−2

x (km)

inlin

e do

wng

oing

fiel

d ((

sJ) 1

/2/m

)Δ r = hs/2Δ r = hsΔ r = 3hs/2

Figure 16. Decomposed downgoing fields using different re-ceiver distributions.

accurately, a dense receiver array is required. We findthat the required receiver separation ∆r must be lessthan the height hs of the dipole source above the seabottom to adequately carry on the up-down decompo-sition (∆r < hs). The following two figures (figures 16and 17) demonstrate the effect on the up-down decom-position of the different receive distributions. The thicksolid lines shows the downgoing (figure 17) and upgo-ing field (figure 16) using a over sampled dense receiverarray (∆r = hs/2). The dashed lines represent decom-posed fields using a receiver spacing of ∆r = hs/2. Theoscillations in the dashed lines imply that this spacingis at the edge of the aliasing. When the receiver sam-pling separation ∆r is larger than hs, the up-down de-

48 Y. Fan & R. Snieder

−10 −5 0 5 10

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

x (km)

inlin

e up

goin

g fie

ld (

(sJ)

1/2

/m)

Δ r = hs/2Δ r = hsΔ r = 3hs/2

Figure 17. Decomposed upgoing fields using different re-ceiver distributions.

composition process is aliased and the field can not bedecomposed accurately (thin solid lines).

This sampling criterion, however, is not practicalfor two reasons. The first is that in current CSEM sur-veys, the separation of the receivers is much larger thanthis (20 times) in the field survey. The second is thatthere is only a line of receivers (instead of 2D array aswe used) used in current practical cases. In order tomake this technique practical, the requirement of thedense 2D receiver network must be relaxed. This is thetopic of ongoing research.

4 CONCLUSIONS

We have shown that by using the multi-dimensional-deconvolution interferometry in a 3D synthetic CSEMsurvey, the airwave effect in a shallow sea is removed.The reflection response of the subsurface is obtained,which contains information only from blow the receiverlevel. Consequently, the difference between the modelswith and without a high resistive layer is significantlyenlarged. The required receiver sampling criterion, how-ever, is unrealistically dense comparing to the currentCSEM survey. In order to use this technique in a practi-cal survey, we are working on how to relax this receiversampling requiement.

5 ACKNOWLEDGMENT

This work is supported by the Shell GameChanger Pro-gram. Authors thank Evert Slob and Kees Wapenaarfrom Delft University of Technology, Netherlands, forthe helpful discussions. We wish to thank JohannesSinger, Mark Rosenquist, Liam O Suilleabhain, JeffreyJohnson from Shell for the help on the EM modeling.Special thanks go to Jonathan Sheiman, Kurang Mehta

from Shell for their great discussions in the implementa-tion of the theory, and suggestions on code development.

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