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GEOPHYSICS, VOL. 71, NO. 3 �MAY-JUNE 2006�; P. G73–G81, 10 FIGS.10.1190/1.2194513

new formula to compute apparent resistivitiesrom marine magnetometric resistivity data

iuping Chen1 and Douglas W. Oldenburg2

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ABSTRACT

Magnetometric resistivity �MMR� is an electromagnetic�EM� exploration method that has been used successfully toinvestigate electrical-resistivity structures below the sea-floor. Apparent resistivity, derived from the observed azi-muthal component of the magnetic field, often is used as anapproximation to the resistivity of a layered earth. Twocommonly used formulas to compute the apparent resistiv-ity have their own limitations and are invalid for a deep-seaexperiment. In this paper, we derive an apparent-resistivityformula based upon the magnetic field resulting from asemi-infinite electrode buried in a 1D layered earth. Thisnew formula can be applied to both shallow and deep ma-rine MMR surveys. In addition, we address the effects thatarise from the transmitter-receiver �Tx-Rx� depth differenceand the choice of the normalized range �the radial distancebetween transmitter and receiver, divided by the thicknessof seawater� on data interpretation and survey design. Theperformance of the new formula is shown by processingsynthetic and field data.

INTRODUCTION

Among the marine electromagnetic �EM� methods used to in-estigate resistivity structures below the seafloor, the magnetomet-ic resistivity �MMR� method has unique characteristics �Edwardsnd Nabighian, 1991�. The method essentially involves measuringhe magnetic field associated with manmade, noninductive �low-requency or pseudo-dc� current flow energized into the seawaternd seafloor through two vertically separated electrodes �bipole�.he magnetic field measured at the ocean-bottom magnetometerepends upon the total current flow at the seafloor and in the sea-

Manuscript received by the Editor August 27, 2004; revised manuscript1Formerly University of British Columbia, Vancouver, Canada: presently

ond, California 94804. E-mail: jchen16@slb.com.2University of British Columbia, Department of Earth and Ocean

oug@eos.ubc.ca.2006 Society of a Exploration Geophysicists. All rights reserved.

G73

ater. In the presence of an isotropically layered seafloor, the mag-etic field generated by the bipole source possesses an azimuthalymmetry, and the bulk resistivity of the seafloor can be estimatedrom the amplitude of the magnetic field.

Apparent resistivity is a commonly used form to present theeasured field data �Chave et al., 1991�. There are two advantages

f using the apparent resistivity versus Tx-Rx radial distance �orange� curve rather than the magnetic-field sounding curve. First,he azimuthal component B� is always decreasing with increasedange, irrespective of whether the seafloor is more conductive orore resistive at depth. Thus, B� versus range is not a sensitive in-

icator of the resistivity depth variation. From the apparent-re-istivity curve, however, it is much easier to get a sense of the sea-oor structure. Second, in a 1D or multidimensional inversionChen et al., 2002�, we usually are required to provide some back-round resistivity as a reference model to recover superimposedargets. In this regard, the apparent resistivity is a handy tool, pro-iding a first-order approximate background structure to the refer-nce model used in a 1D inversion. The recovered 1D inversionhen may be used as a reference model in any subsequent multidi-

ensional inversions.There are two formulas in the geophysical literature for comput-

ng the apparent resistivity �a from the measured B�. The first isiven in Chave et al. �1991� and provides

�a =�0IH�0

4�R2B�

, �1�

here B� is the measured azimuthal magnetic field at the receiver,is the radial distance between the Rx and Tx wire, �0 is the per-eability of free space and nonmagnetic seafloor, I is the current

trength in the transmitter wire, H denotes the thickness of the sea-ater, and �0 is the resistivity of the seawater, which is presumablynown. Two assumptions are required in the derivation of thisquation: First, the range R must be large compared to the sea

d October 6, 2005; published online May 19, 2006.berger-EMI Technology Center, 1301 South 46 Street, Building 300, Rich-

s, 6339 Stores Road, Vancouver, British Columbia, Canada. E-mail:

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G74 Chen and Oldenburg

epth H; second, the integrated conductivity of the sea layer �0Hust be large compared with the parameter �1R, where �1 is sea-oor conductivity �i.e., H � R � �0/�1H�. These assumptions areecessary so that the bipole current is channeled out to relativelyarge distances by the sea. Typical values of conductivities are �0

3.3 S/m and �1 = 0.01 to 2.0 S/m. Therefore, the Chave et al.1991� method for calculating apparent resistivity works best in

shallow ocean, as in the MOSES experiment �Edwards et al.,985�, where H = 200 m. However, in a deep-sea survey, R/Hanges from 0.1–3.0 �Evans et al., 1998; Evans et al., 2002�. As aesult, the first assumption fails, and the formula will not provide aood approximation �see Figure 1�. Clearly, the apparent-resist-vity curve provides no indication of the layered-resistivity struc-ure of the seafloor, especially the deep, low-resistivity zone.

The second formula is given in Wolfgram et al. �1986�, where �a

s obtained by

�a =�0I�0

2�RB�

H�H2 + R2

− �0. �2�

he formula ignores the effect from the electrode on the sea sur-ace by assuming that the top electrode is located at infinity. To de-ive this equation, R must be smaller than H �i.e., R � H�. This as-umption limits the formula’s use in deep-sea MMR because thex-Rx separation R can be greater than H. As Figure 1b illustrates,

he Wolfgram et al. �1986� method offers a poor indication ofhree-layer structure. When R/H is small �0.1 in this example�, thepparent resistivity approaches the true value �7 �.m�; otherwise,he formula provides an inadequate approximation.

DERIVING A NEW APPARENT RESISTIVITY

To derive a general apparent-resistivity formula for a marineMR survey, we need an analytic or semianalytic expression for

he magnetic field resulting from a semi-infinite wire source with aoint electrode buried in a layered seafloor.

agnetic field resulting from aemi-infinite source in a 1D earth

As shown in Figure 2, a semi-infinite vertical wire AOC carriesn excitation current I and terminates at the location C. The elec-

igure 1. �a� A three-layer seafloor resistivity model in a deep mararent resistivities versus normalized radial Tx-Rx distance �rangel. �1991�, Wolfgram et al. �1986�, and our new formulas.

rode C is placed at the interface z = zs between layers s and s + 1o simplify the mathematics. Each layer has a constant conductiv-ty � j with thickness hj and a magnetic permeability equal to freepace. There are a total of N − 1 interfaces, with �N as the termi-ating half-space. In the source-free region, the magnetic field Bbeys

� �1

�� � B = 0 . �3�

The problem is axisymmetric, and B has only an azimuthal com-onent in cylindrical coordinates r,�,z. For simplicity, we use B toepresent the azimuthal component in the following derivations.xpanding equation 3 and neglecting the conductivity, because �

s a constant in each layer, yields

�2B

�r2 +1

r

�B

�r−

1

r2B +�2B

�z2 = 0. �4�

Following the Hankel transform method �Edwards and Nabig-ian, 1991�, we define a Hankel transform pair as

B��,z� = �0

rB�r,z�J1��r�dr �5�

nd

B�r,z� = �0

�B��,z�J1��r�d� , �6�

here J1 is the Bessel function of the first kind of order one. Theankel transformation of equation 4 results in the simple, second-rder equation in the wavenumber domain �

�2B

�z2 − �2B = 0, �7�

here B is the magnetic field in the wavenumber domain. Acomplementary solution to equation 7 inany layer j is

Bj��,z� = Dje−��z−zj−1� + Uje

��z−zj−1�,

�8�

where Dj and Uj are the downward- andupward-propagation coefficients, inde-pendent of the variable z but dependenton conductivity �. The values Dj and Uj

can be determined through a propagatormatrix by applying boundary conditionsat the layer interfaces. To determine Dj

and Uj, we use boundary conditions inwhich the azimuthal component of mag-netic field B and the radial component ErMR survey. �b� Ap-

ned using Chave et

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Apparent resistivity for marine MMR G75

f the electric field are continuous across the interface, i.e.,

Bj���,z��z=zj= Bj+1���,z��z=zj

�9�

nd

Ejr���,z��z=zj

= Ej+1r ���,z��z=zj

. �10�

n addition, we have the constraints

UN = 0, �11�

.e., there are no upcoming fields in the last layer, and

��Ez�z=0 = 0. �12�

onstraint 12 requires that there is no current crossing the air-eawater interface. Therefore, the 2N unknown coefficients can beetermined from the 2�N − 1� + 2 equations. Further details cane found in Chen and Oldenburg �2004�.

pparent-resistivity formulas

Suppose the Tx bipole extends from the sea surface to the seaottom �length L = H� and the seafloor is a uniform half-spaceith resistivity �1 �see Figure 3�. Using the wavenumber method

bove, we can compute the magnetic field at the seafloor �depth H�esulting from one semi-infinite wire terminating at the sea surfacethe top electrode is assumed negative� in the wavenumber do-ain,

B−��,H� = −�0I

4��

4�0

�1e−�H

�1 +�0

�1� − �1 −

�0

�1�e−2�H

, �13�

nd another semi-infinite wire terminating at the seafloor �the bot-om positive electrode�,

igure 2. Schematic of a semi-infinite wire source buried in a lay-red earth.

B+��,H� =�0I

2��

�0

�1�1 + e−2�H�

�1 +�0

�1� − �1 −

�0

�1�e−2�H

. �14�

he total field resulting from the bipole is B��,H� = B+ + B−. Theenominator term in both equations can be expanded in a binomialpproximation as

1

�1 +�0

�1� − �1 −

�0

�1�e−2�H

=1

�1 +�0

�1��1 +

�1 −�0

�1�

�1 +�0

�1� e−2�H + HOT , �15�

here HOT stands for higher-order terms of e−2�H. Substitutingquation 15 into B+ and B− yields

B��,H� =�0I

2��

�0

�1

1 +�0

�1

n=0

ne−n�H, �16�

here

n = �1 n = 0

− 2�m−1 n = 2m − 1

�m−1�1 + �� n = 2m, m = 1,2, . . .� �17�

nd

� =

1 −�0

�1

1 +�0

�1

. �18�

igure 3. A two-layer �including seawater� model for defining ap-arent resistivity.

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G76 Chen and Oldenburg

ransforming back to the spatial domain and making use of the in-egral identity

�0

e−n�HJ1��R�d� =1

R 1 −nH

�R2 + �nH�2� �19�

ields

B�R,H� =�0I

2�R

�0

�0 + �1�1 + n=1

n�1 −1

�1 + � R

nH�2� .

�20�o obtain a simple approximate relation between B and �0/�1, weefine F�R/H,�0/�1�, which is a function of R/H and �0/�1, to rep-esent the content within the brackets:

F� R

H,�0

�1� = 1 +

n=1

n�1 −1

�1 + � R

nH�2 . �21�

unction F�R/H,�0/�1� is displayed in Figure 4, where R/H rangesrom 0.01 to 10 while the ratio �0/�1 is 0.001, 0.01, 0.1, and 0.5, re-pectively. When R/H � 0.2, the function F is independent of0/�1. In addition, when �0/�1 � 0.01, F depends only on R/H. Weill take advantage of this feature to develop a simple relationship

n the following derivation.One approach to obtain an approximate form of equation 21 is to

runcate the infinite series in that equation at some value of n andepresent the result as Fn�R/H,�0/�1�. For example,

F1� R

H,�0

�1� =

2

�1 + � R

H�2

− 1, �22�

igure 4. Infinite series function F�R/H,�0/�1� changes with R/Hor different � /� ratios.

0 1

F2� R

H,�0

�1� =

2

�1 + � R

H�2

−1 + �

�1 + � R

2H�2

+ � ,

�23�

nd so on. Unfortunately, this is not a good approach because ofhe oscillating nature of the coefficients n, illustrated in Figure 5,here ratio �0/�1 = 0.01 is used. Interestingly, if we look at F1 in

quation 22, we find that the Wolfgram et al. �1986� formula ig-ores �1/�1 + �R/H�2� − 1; in other words,

Fw� R

H,�0

�1� =

1

�1 + � R

H�2

, �24�

hich does a better job than F1 to approximate to the infinite se-ies.

Following that insight, we begin to develop a formula using twoerms in the series expansion �n = 2�. As we note from Figure 5, F2

s not a good approximation, but when we delete terms � − ��/1 + �R/�2H��2� from equation 23, the remainder performs better

n terms of getting closer to the true F when R/H increases. As aurther modification, we replace unity by in the second term, sour expression has the form

F � R

H,�0

�1� =

2

�1 + � R

H�2

−

�1 + � R

2H�2

. �25�

he unknown , a function of R/H and �0/�1, can be obtained bytting to the curves shown in Figure 4. To conveniently pick an alue, Figure 6a shows the lookup curve of versus R/H and0/�1, and Figure 6b is a contour map for . Alternatively, can beomputed through an explicit expression

igure 5. Four truncated functions Fn�R/H,�0/�1� to approximatehe infinite series function �the true function F�. A resistivity ratio

/� = 0.01 was used.

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= �1.0 ifR

H� 0.2

�x,y�, x = log10� R

H�, y = log10��0

�1� otherwise � ,

�26�

here

�x,y� = �− 0.023 − 0.055y + 0.101y2� + �− 0.012

− 0.029y + 0.005y2 + 0.087y3�x + �0.023

+ 0.080y − 0.124y2�x2 + �0.021 + 0.064y

− 0.136y2�x3 + �0.005 + 0.016y + 0.964y2�x4.

�27�

ll of the coefficients are obtained by fitting a polynomial of orderour in x and order three in y in a least-squares sense.

In general, we do not know exactly what �0/�1 is; fortunately, the

igure 6. Alphas can be determined from �a� lookup curve or �b�ontour map for different R/H and �0/�1. Note = 1 when R/H

0.2 in �b�. The grayscale bar for is unitless.

alue of is not extremely sensitive to �0/�1. From Figure 6a, evenhere �0/�1 varies almost three decades �0.001 to 0.5�, only

hanges in the range of 0.75–1.0. This means that even a poor esti-ation of �0/�1 will not make a significant impact on selecting anfrom the lookup curve. In this sense, determination of in equa-

ion 26 is robust and stable. Finding a good truncated function F ,e can define the corresponding apparent resistivity by

�a =�0I

2�R

�0

B�� 2

�1 + � R

H�2

−

�1 + R

�2H��2� − �0.

�28�

Equation 28 is actually the simplest situation encountered in aractical survey. Because of the bathymetry of the ocean bottom,he lower electrode of the transmitter might be located at a depthifferent from the depth of the receiver. A general model can beresented by locating both the lower electrode and the receiver atifferent depths in the seawater. Depending upon the relativeepth, we consider the problem in two cases, as shown in Figure 7.

ase A

In this case, the magnetometer is located at depth Zr�Zr � LH�, simulating the situation in which the transmitter is near the

ea bottom while the receiver is at a hill because of bathymetry ofhe seafloor. Generally, we follow the same procedure as above toerive the magnetic field. However, this derivation is more compli-ated because we have two additional depths, Zr and L, and haveore combinations among Zr, L, and H. More importantly, truncat-

ng the infinite series in the spatial domain has proven unsatis-actory because of its oscillating behavior. We have resorted to alightly different method to find an optimum �R/H,�0/�1� in thisase. First, we truncate the infinite series directly in the wavenum-er domain and retain exponential terms up to n = 2 for the mag-etic field �e.g., we only have exponential terms such as e−�H, e−2�H,−�Zr, e−��2H−Zr�, etc.�. Second, we transform these related terms intohe spatial domain using the integral identity �equation 19�. Finally,ssembling them yields the total magnetic field

B�R,Zr� =�0I

4�R

�1

�0 + �1 �0

�1A1 + A2� , �29�

here the coefficients are

igure 7. Two general scenarios for Tx-Rx configuration in marineMR: �a� Z � L � H and �b� L � Z � H.

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G78 Chen and Oldenburg

A1 =2Zr

�R2 + Zr2

+2�2H − Zr�

�R2 + �2H − Zr�2

+2H + Zr − L

�R2 + �2H + Zr − L�2+

L − Zr

�R2 + �L − Zr�2

−4H

�R2 + �2H�2−

Zr + L�R2 + �Zr + L�2

−2H − Zr − L

�R2 + �2H − Zr − L�2�30�

nd

A2 =2Zr

�R2 + Zr2

+2H − Zr − L

�R2 + �2H − Zr − L�2+

4H�R2 + �2H�2

+L − Zr

�R2 + �L − Zr�2−

2�2H − Zr��R2 + �2H − Zr�2

−Zr + L

�R2 + �Zr + L�2−

2H + Zr − L�R2 + �2H + Zr − L�2

. �31�

ccordingly, the apparent resistivity can be defined as

�a =�0I

4�R

�0

B�

1

A3A1 −

�0

A3, �32�

here

A3 = 1 −�0IA2

4�RB�

. �33�

f Zr = L = H, then A1 = 4H/�R2 + H2 − 4H/�R2 + �2H�2, A2 = 0,3 = 1, and equation 32 is identical to equation 28. Since similarssumptions also are made in the derivation, we must correct forquation 28 with = 1. This means A1 can be modified as

A1 =2

�1 + � R

Zr�2

+2

�1 + R

�2H − Zr��2

+

�1 + R

�2H + Zr − L��2+

L − Zr

�R2 + �L − Zr�2

−2

�1 + R

�2H��2−

�1 + R

�Zr + L��2

−2H − Zr − L

�R2 + �2H − Zr − L�2, �34�

here can be approximately determined by equation 26. The ef-ect of on A and A is very small and can be neglected.

2 3

ase B

In this case, the magnetometer might be below the lower elec-rode �L � Zr � H�, simulating the situation in which the trans-itter wire is seated at a hill while the receiver is at a valley. Simi-

ar to case A, the total azimuthal field is

B�R,Zr� =�0I

4�R

�1

�0 + �1 �0

�1A4 + A5� , �35�

here the coefficients are

A4 =2Zr

�R2 + Zr2

+2�2H − Zr�

�R2 + �2H − Zr�2−

Zr − L�R2 + �Zr − L�2

−Zr + L

�R2 + �Zr + L�2−

2H + L − Zr

�R2 + �2H + L − Zr�2

−2H − Zr − L

�R2 + �2H − Zr − L�2�36�

nd

A5 =2Zr

�R2 + Zr2

+2H + L − Zr

�R2 + �2H + L − Zr�2

+2H − L − Zr

�R2 + �2H − L − Zr�2−

L + Zr

�R2 + �L + Zr�2

−2�2H − Zr�

�R2 + �2H − Zr�2−

Zr − L�R2 + �Zr − L�2

. �37�

he apparent resistivity can be expressed as

�a =�0I

4�R

�0

B�

1

A6A4 −

�0

A6, �38�

here

A6 = 1 −�0IA5

4�RB�

. �39�

fter correction, coefficient A4 reads

A4 =2

�1 + � R

Zr�2

+2

�1 + R

�2H − Zr��2

−

�1 + R

�L + Zr��2

−

�1 + R

�2H + L − Zr��2

−Zr − L

�R2 + �Z − L�2−

2H − Zr − L�R2 + �2H − Z − L�2

. �40�

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Apparent resistivity for marine MMR G79

s a consistency check, when Zr = L � H, cases A and B shoulde identical. In other words, we will have A1 � A4 and A2 � A5.his is true for our derivations.

erification

As a verification, we use the new formula derived in equation 28o compute the apparent resistivity for the three-layer model shownn Figure 1a. We assume �0/�1 = 0.01, and corresponding values of

are computed from equation 26. The new sounding curve cor-ectly reveals the three-layer structure and gives a good approxi-ation to both the first layer and basement resistivity. More impor-

antly, there is no restriction on the normalized distance R/H. Theew formula works over a wider range �0.01 � R/H � 10�.

APPLICATIONS

The new apparent-resistivity formula provides a useful tool toddress some practical issues that arise in a marine MMR survey.irst, we look at the effect of the relative vertical offset between

he transmitter and the receiver. We then show that it is necessaryo acquire data over a large range of R/H. Finally, we apply the de-ived formula to field data from the East Pacific Rise.

ffect of Tx-Rx vertical offset

Although marine MMR surveys haveeen conducted in several places �Ed-ards et al., 1985; Evans et al., 1998;vans et al., 2002�, we feel there are stillome important practical questions to an-wer in order to apply this method moreffectively. For simplicity of data pro-essing and interpretation, it usually is as-umed that the transmitter and receiverre located at the same depth below theea surface. In practice, receivers are al-ays dropped on the seafloor. However,

he transmitter wire is often hanging inhe seawater, not in contact with the sea-oor, or the transmitter and receiver are

ocated at different depths because of theathymetry of the seafloor. For example,he receiver might be on the ridge crestith the transmitter deployed at a deeperepth. If the vertical distance between theower end of the transmitter wire and the

agnetometer is much less than thehickness of the seawater �e.g., the ratio is%�, the geometric difference may beegligible. However, if the ratio is ap-roximately 10%, then the difference hassignificant effect on the measured mag-etic fields. Without taking the geometricifference into account, the interpretationill be compromised. A synthetic ex-

mple is presented in Figure 8.Figure 8c assumes a layered model

here the resistivities of the seawater andeafloor are 0.3 and 10 �.m, respectively.

Figure 8. Effect of thand �b� the derived a

he transmitters are located 2700 m below the sea surface, whilehe magnetometer is at 2500 m depth. This simulates the casehere the magnetometer is situated on the ridge axis without tak-

ng the bathymetry into account. Figure 8a shows the azimuthal B�

ersus the normalized distance R/H for the on-axis magnetometer.or comparison, the off-axis magnetic field is plotted also. Surpris-

ngly, the amplitudes for the on-axis receiver are much largerabout one order of magnitude� than those for the off-axis receiver.his significant difference results purely from the vertical shift of

eceiver location. Without taking the transmitter-receiver geomet-ic difference into account, the derived resistivities of the on-axisesponse varies from 0.2–15 �.m, while the off-axis responseields values from 10–18 �.m �solid dots and circles in Figure 8b,btained using Wolfgram et al., 1986�. Obviously, these results arensatisfactory. When we use the new formula to obtain the appar-nt resistivity, both curves �solid and dashed lines in Figure 8b� of-er a good approximation to the model value �10 �.m�.

ffect of normalized range

Analysis of the apparent-resistivity curve reveals the importancef the normalized distance on the data interpretation and survey de-

xis and off-axis magnetometers on �a� the observed magnetic fieldst resistivity. �c� The 1D model used to investigate the effect.

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G80 Chen and Oldenburg

ign. As shown in Figure 9a, the apparent-resistivity curve ob-ained with a normalized range 0.04 � R/H � 4 �labeled full

igure 9. Effect of the normalized range on data inversion and survistivities for the three-layer seafloor model. �b� Recovered 1D resiR/H = 0.3–4� and a full data range �0.04–4�.

ange� clearly shows a three-layer model of the seafloor. The insets the true 1D model. When we carry out a 1D inversion based

upon a generalized cross-validation tech-nique with full-range data, the recoveredstructure reveals the lower-resistivitylayer in a three-layer model �see Figure9b�. Conversely, if we only use the data inthe normalized range 0.3 � R/H � 4 �la-beled small range�, the recovered modelfrom a 1D inversion indicates a two-layerstructure. While this can be explainedeasily from the apparent-resistivity curve,it is not as obvious if we look at themagnetic-field curve. This simple exam-ple suggests that if the normalized dis-tance is not covered widely enough, wewill likely miss the shallow-resistivity in-formation, resulting in a poor 1D model.In this regard, choice of the normalizedrange has a definitive impact on the sur-vey design.ign. �a� apparent re-

s with a small range

igure 10. Apparent-resistivity map associated with instrumentve at EPR using �a� the Wolfgram et al. �1986� formula and �b�ur formula. �c� Comparison of the apparent resistivity along thex 5-A profile marked in �a� and �b�. The large red dot is the loca-

ion of Rx instrument five. Note that the apparent resistivity com-uted for each transmitter is plotted at the transmitter location. Theolor scales �in �.m� are slightly different, for it is difficult toake them identical.

ey desstivitie

FfioRtpcm

D

Pctegdtwtg1trtOWmftat

tsmidcdmtbam

IAbnCCgvfMcv

C

C

C

E

E

E

E

W

Apparent resistivity for marine MMR G81

ata set

A marine MMR experiment recently was conducted at the Eastacific Rise �EPR� to study the electrical resistivity of the shallowrust in the vicinity of the ridge �Evans et al., 2002�. More than 200ransmitter bipoles and 10 magnetometers were deployed in thisxperiment. The magnetometers could be categorized into tworoups: on axis and off axis. For the on axis magnetometers, theepth of the receiver was approximately 2500 m; the depth for theransmitters varied from 2500–2700 m. We chose instrument five,hich was on the crest of the ridge, and its associated 160 transmit-

ers as an example. The apparent resistivities obtained using Wolf-ram et al. �1986� and our formulas are shown in Figures 10a and0b. There are some similarities in these two plots. It appears thathe shallow material has low resistivity, and deep material has aelatively increasing resistivity. The difference can be seen fromhe apparent-resistivity profile along receiver 5-A �see Figure 10c�.ur curve shows a resistivity low between 1.5 and 4.5 km, whileolfgram et al. �1986� does not. This resistivity low suggests thereay be a fairly low-resistive layer at depth. Because of limited in-

ormation, we cannot make the judgment that our result is betterhan that of Wolfgram et al. �1986� for this example. More work on3D inversion must be carried out to obtain a 3D electrical struc-

ure in this region.

CONCLUSION

We have derived a new apparent-resistivity formula based uponhe semianalytic expression for the magnetic field resulting from aemi-infinite electrode source buried in a 1D earth. The new for-ula is superior to the two most commonly used formulas in that it

s accurate for a full range of the normalized transmitter-receiveristance. We have also investigated the effects of transmitter-re-eiver geometric difference and the choice of normalized range onata interpretation and survey design. The utility of the derived for-ula is demonstrated with synthetic and field data sets. We believe

hat first-order approximate resistivity information can be obtainedy converting the observed magnetic field to apparent resistivity,nd that this can assist data interpretation, survey design, and esti-

ation of a background model for 3D inversion.

ACKNOWLEDGMENTS

The work presented here was funded by NSERC and theMAGE Consortium, of which the following are members: AGIP,nglo American Corporation, BHP Billiton, EMI Inc., Falcon-ridge Ltd., INCO Exploration and Technical Services Inc., Ken-ecott Exploration, MIM Exploration Party Ltd., Muskox Mineralsorp., Newmont Exploration Ltd., Placer Dome Inc., and Teckominco Ltd. We are grateful for their participation. Thanks alsoo to Robert Evans of Woods Hole Oceanography Institute for pro-iding the East Pacific Rise data and marine MMR background in-ormation. The editing staff, including Jerry Schuster, Yonghe Sun,

ark Everett, and three anonymous reviewers, offered insightfulomments on the original manuscript, prompting a much-refinedersion. We are indebted for their help.

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