A new formula to compute apparent resistivities from marine magnetometric resistivity data

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1 GEOPHYSICS, VOL. 71, NO. 3 MAY-JUNE 006; P. G73 G81, 10 FIGS / A new formula to compute apparent resistivities from marine magnetometric resistivity data Jiuping Chen 1 Douglas W. Oldenburg ABSTACT Magnetometric resistivity MM is an electromagnetic EM exploration method that has been used successfully to investigate electrical-resistivity structures below the seafloor. Apparent resistivity, derived from the observed azimuthal component of the magnetic field, often is used as an approximation to the resistivity of a layered earth. Two commonly used formulas to compute the apparent resistivity have their own limitations are invalid for a deep-sea experiment. In this paper, we derive an apparent-resistivity formula based upon the magnetic field resulting from a semi-infinite electrode buried in a 1D layered earth. This new formula can be applied to both shallow deep marine MM surveys. In addition, we address the effects that arise from the transmitter-receiver Tx-x depth difference the choice of the normalized range the radial distance between transmitter receiver, divided by the thickness of seawater on data interpretation survey design. The performance of the new formula is shown by processing synthetic field data. INTODUCTION Among the marine electromagnetic EM methods used to investigate resistivity structures below the seafloor, the magnetometric resistivity MM method has unique characteristics Edwards Nabighian, The method essentially involves measuring the magnetic field associated with manmade, noninductive lowfrequency or pseudo-dc current flow energized into the seawater seafloor through two vertically separated electrodes bipole. The magnetic field measured at the ocean-bottom magnetometer depends upon the total current flow at the seafloor in the seawater. In the presence of an isotropically layered seafloor, the magnetic field generated by the bipole source possesses an azimuthal symmetry, the bulk resistivity of the seafloor can be estimated from the amplitude of the magnetic field. Apparent resistivity is a commonly used form to present the measured field data Chave et al., There are two advantages of using the apparent resistivity versus Tx-x radial distance or range curve rather than the magnetic-field sounding curve. First, the azimuthal component B is always decreasing with increased range, irrespective of whether the seafloor is more conductive or more resistive at depth. Thus, B versus range is not a sensitive indicator of the resistivity depth variation. From the apparent-resistivity curve, however, it is much easier to get a sense of the seafloor structure. Second, in a 1D or multidimensional inversion Chen et al., 00, we usually are required to provide some background resistivity as a reference model to recover superimposed targets. In this regard, the apparent resistivity is a hy tool, providing a first-order approximate background structure to the reference model used in a 1D inversion. The recovered 1D inversion then may be used as a reference model in any subsequent multidimensional inversions. There are two formulas in the geophysical literature for computing the apparent resistivity a from the measured B. The first is given in Chave et al provides a = 0IH 0 4 B, where B is the measured azimuthal magnetic field at the receiver, is the radial distance between the x Tx wire, 0 is the permeability of free space nonmagnetic seafloor, I is the current strength in the transmitter wire, H denotes the thickness of the seawater, 0 is the resistivity of the seawater, which is presumably known. Two assumptions are required in the derivation of this equation: First, the range must be large compared to the sea 1 Manuscript received by the Editor August 7, 004; revised manuscript received October 6, 005; published online May 19, Formerly University of British Columbia, Vancouver, Canada: presently Schlumberger-EMI Technology Center, 1301 South 46 Street, Building 300, ichmond, California jchen16@slb.com. University of British Columbia, Department of Earth Ocean Sciences, 6339 Stores oad, Vancouver, British Columbia, Canada. doug@eos.ubc.ca. 006 Society of a Exploration Geophysicists. All rights reserved. G73

2 G74 Chen Oldenburg depth H; second, the integrated conductivity of the sea layer 0 H must be large compared with the parameter, where is seafloor conductivity i.e., H 0 / H. These assumptions are necessary so that the bipole current is channeled out to relatively large distances by the sea. Typical values of conductivities are 0 = 3.3 S/m = 0.01 to.0 S/m. Therefore, the Chave et al method for calculating apparent resistivity works best in a shallow ocean, as in the MOSES experiment Edwards et al., 1985, where H = 00 m. However, in a deep-sea survey, /H ranges from Evans et al., 1998; Evans et al., 00.Asa result, the first assumption fails, the formula will not provide a good approximation see Figure 1. Clearly, the apparent-resistivity curve provides no indication of the layered-resistivity structure of the seafloor, especially the deep, low-resistivity zone. The second formula is given in Wolfgram et al. 1986, where a is obtained by a = 0I 0 H B H + 0. The formula ignores the effect from the electrode on the sea surface by assuming that the top electrode is located at infinity. To derive this equation, must be smaller than H i.e., H. This assumption limits the formula s use in deep-sea MM because the Tx-x separation can be greater than H. As Figure 1b illustrates, the Wolfgram et al method offers a poor indication of three-layer structure. When /H is small 0.1 in this example, the apparent resistivity approaches the true value 7.m; otherwise, the formula provides an inadequate approximation. DEIVING A NEW APPAENT ESISTIVITY To derive a general apparent-resistivity formula for a marine MM survey, we need an analytic or semianalytic expression for the magnetic field resulting from a semi-infinite wire source with a point electrode buried in a layered seafloor. Magnetic field resulting from a semi-infinite source in a 1D earth As shown in Figure, a semi-infinite vertical wire AOC carries an excitation current I terminates at the location C. The electrode C is placed at the interface z = z s between layers s s +1 to simplify the mathematics. Each layer has a constant conductivity j with thickness h j a magnetic permeability equal to free space. There are a total of N 1 interfaces, with N as the terminating half-space. In the source-free region, the magnetic field B obeys B = 0. 3 The problem is axisymmetric, B has only an azimuthal component in cylindrical coordinates r,,z. For simplicity, we use B to represent the azimuthal component in the following derivations. Exping equation 3 neglecting the conductivity, because is a constant in each layer, yields B r + 1 B r r 1 r B + B z =0. Following the Hankel transform method Edwards Nabighian, 1991, we define a Hankel transform pair as B,z = 0 Br,z =0 rbr,zj 1 rdr B,zJ 1 rd, where J 1 is the Bessel function of the first kind of order one. The Hankel transformation of equation 4 results in the simple, secondorder equation in the wavenumber domain B z B =0, where B is the magnetic field in the wavenumber domain. A complementary solution to equation 7 in any layer j is B j,z = D je zz j1 + Ũ j e zz j1, 8 Figure 1. a A three-layer seafloor resistivity model in a deep marine MM survey. b Apparent resistivities versus normalized radial Tx-x distance range obtained using Chave et al. 1991, Wolfgram et al. 1986, our new formulas. where D j Ũ j are the downward- upward-propagation coefficients, independent of the variable z but dependent on conductivity. The values D j Ũ j can be determined through a propagator matrix by applying boundary conditions at the layer interfaces. To determine D j Ũ j, we use boundary conditions in which the azimuthal component of magnetic field B the radial component E r

3 Apparent resistivity for marine MM G75 of the electric field are continuous across the interface, i.e., B j,z z=zj = B j+1,z z=zj Ẽ r r j,z z=zj = Ẽ j+1,z z=zj. In addition, we have the constraints 9 10 B +,H = 0I 0 1+e H e H. 14 The total field resulting from the bipole is B,H = B + + B. The denominator term in both equations can be exped in a binomial approximation as 1 Ũ N =0, i.e., there are no upcoming fields in the last layer, E z z=0 = Constraint 1 requires that there is no current crossing the airseawater interface. Therefore, the N unknown coefficients can be determined from the N 1 + equations. Further details can be found in Chen Oldenburg e H = eh HOT, 15 where HOT sts for higher-order terms of e H. Substituting equation 15 into B + B yields Apparent-resistivity formulas Suppose the Tx bipole extends from the sea surface to the sea bottom length L = H the seafloor is a uniform half-space with resistivity see Figure 3. Using the wavenumber method above, we can compute the magnetic field at the seafloor depth H resulting from one semi-infinite wire terminating at the sea surface the top electrode is assumed negative in the wavenumber domain, B,H = 0I e H e H, 13 where 0 0 I B,H = 1+ n e nh, 0 n=0 16 n =0 n = m1 n =m 1 17 m1 1+ n =m, m = 1,,... another semi-infinite wire terminating at the seafloor the bottom positive electrode, = Figure. Schematic of a semi-infinite wire source buried in a layered earth. Figure 3. A two-layer including seawater model for defining apparent resistivity.

4 G76 Chen Oldenburg Transforming back to the spatial domain making use of the integral identity yields 0 e nh J 1 d = B,H = 0I 0 + n=1 nh + nh 19 n 1 nh. 0 To obtain a simple approximate relation between B 0 /,we define F/H, 0 /, which is a function of /H 0 /, to represent the content within the brackets: F H, 0 =1+ n=1 nh 1 Function F/H, 0 / is displayed in Figure 4, where /H ranges from 0.01 to 10 while the ratio 0 / is 0.001, 0.01, 0.1, 0.5, respectively. When /H 0., the function F is independent of 0 /. In addition, when 0 / 0.01, F depends only on /H.We will take advantage of this feature to develop a simple relationship in the following derivation. One approach to obtain an approximate form of equation 1 is to truncate the infinite series in that equation at some value of n represent the result as F n /H, 0 /. For example, F 1 H, 0 = n 1. H 1, F H, 0 = H 1+ H +, 3 so on. Unfortunately, this is not a good approach because of the oscillating nature of the coefficients n, illustrated in Figure 5, where ratio 0 / = 0.01 is used. Interestingly, if we look at F 1 in equation, we find that the Wolfgram et al formula ignores 1/+/H 1; in other words, F w H, 0 = 1 H, 4 which does a better job than F 1 to approximate to the infinite series. Following that insight, we begin to develop a formula using two terms in the series expansion n =. As we note from Figure 5, F is not a good approximation, but when we delete terms / +/H from equation 3, the remainder performs better in terms of getting closer to the true F when /H increases. As a further modification, we replace unity by in the second term, so our expression has the form F H, 0 = H H. 5 The unknown, a function of /H 0 /, can be obtained by fitting to the curves shown in Figure 4. To conveniently pick an value, Figure 6a shows the lookup curve of versus /H 0 /, Figure 6b is a contour map for. Alternatively, can be computed through an explicit expression Figure 4. Infinite series function F/H, 0 / changes with /H for different 0 / ratios. Figure 5. Four truncated functions F n /H, 0 / to approximate the infinite series function the true function F. A resistivity ratio 0 / = 0.01 was used.

5 Apparent resistivity for marine MM G77 =1.0 where if 0. H x,y, x = log 10 H, y = log 10 0 otherwise, 6 x,y = y y y y y 3 x y 0.14y x y 0.136y x y y x 4. 7 All of the coefficients are obtained by fitting a polynomial of order four in x order three in y in a least-squares sense. In general, we do not know exactly what 0 / is; fortunately, the value of is not extremely sensitive to 0 /. From Figure 6a, even where 0 / varies almost three decades to 0.5, only changes in the range of This means that even a poor estimation of 0 / will not make a significant impact on selecting an from the lookup curve. In this sense, determination of in equation 6 is robust stable. Finding a good truncated function F, we can define the corresponding apparent resistivity by a = 0I 0 B H H 0. 8 Equation 8 is actually the simplest situation encountered in a practical survey. Because of the bathymetry of the ocean bottom, the lower electrode of the transmitter might be located at a depth different from the depth of the receiver. A general model can be presented by locating both the lower electrode the receiver at different depths in the seawater. Depending upon the relative depth, we consider the problem in two cases, as shown in Figure 7. Case A In this case, the magnetometer is located at depth Z r Z r L H, simulating the situation in which the transmitter is near the sea bottom while the receiver is at a hill because of bathymetry of the seafloor. Generally, we follow the same procedure as above to derive the magnetic field. However, this derivation is more complicated because we have two additional depths, Z r L, have more combinations among Z r, L, H. More importantly, truncating the infinite series in the spatial domain has proven unsatisfactory because of its oscillating behavior. We have resorted to a slightly different method to find an optimum /H, 0 / in this case. First, we truncate the infinite series directly in the wavenumber domain retain exponential terms up to n = for the magnetic field e.g., we only have exponential terms such as e H, e H, e Z r, e HZ r, etc.. Second, we transform these related terms into the spatial domain using the integral identity equation 19. Finally, assembling them yields the total magnetic field B,Z r = 0I A 1 + A, 9 where the coefficients are Figure 6. Alphas can be determined from a lookup curve or b contour map for different /H 0 /. Note = 1 when /H 0. in b. The grayscale bar for is unitless. Figure 7. Two general scenarios for Tx-x configuration in marine MM: a Z r L H b L Z r H.

6 G78 Chen Oldenburg A 1 = Z r + Z r + + H Z r + H Z r H + Z r L + H + Z r L + L Z r + L Z r Case B In this case, the magnetometer might be below the lower electrode L Z r H, simulating the situation in which the transmitter wire is seated at a hill while the receiver is at a valley. Similar to case A, the total azimuthal field is 4H + H H Z r L + H Z r L Z r + L + Z r + L 30 B,Z r = 0I A 4 + A 5, where the coefficients are 35 A = Z r + Z r + + L Z r + L Z r Z r + L + Z r + L H Z r L + H Z r L + H Z r + H Z r 4H + H H + Z r L + H + Z r L. Accordingly, the apparent resistivity can be defined as where a = 0I 0 1 A 1 0, 4 B A 3 A 3 A 3 =1 0IA 4B If Z r = L = H, then A 1 =4H/ + H 4H/ + H, A =0, A 3 = 1, equation 3 is identical to equation 8. Since similar assumptions also are made in the derivation, we must correct for equation 8 with = 1. This means A 1 can be modified as A 1 = Z r + H Z r + H + Z r L + L Z r + L Z r H Z r + L H Z r L + H Z r L, 34 where can be approximately determined by equation 6. The effect of on A A 3 is very small can be neglected. A 4 = Z r + Z r + A 5 = Z r + L + Z r + L H Z r + H Z r H Z r L + H Z r L Z r + Z r + + Z r L + Z r L H + L Z r + H + L Z r H + L Z r + H + L Z r H L Z r + H L Z r H Z r + H Z r The apparent resistivity can be expressed as where L + Z r + L + Z r Z r L + Z r L. a = 0I 0 1 A 4 0, 4 B A 6 A 6 A 6 =1 0IA 5 4B. After correction, coefficient A 4 reads A 4 = Z r + L + Z r Z r L + Z r L H Z r H + L Z r H Z r L + H Z r L

7 Apparent resistivity for marine MM G79 As a consistency check, when Z r = L H, cases A B should be identical. In other words, we will have A 1 A 4 A A 5. This is true for our derivations. Verification As a verification, we use the new formula derived in equation 8 to compute the apparent resistivity for the three-layer model shown in Figure 1a. We assume 0 / = 0.01, corresponding values of are computed from equation 6. The new sounding curve correctly reveals the three-layer structure gives a good approximation to both the first layer basement resistivity. More importantly, there is no restriction on the normalized distance /H. The new formula works over a wider range 0.01 /H 0. APPLICATIONS The new apparent-resistivity formula provides a useful tool to address some practical issues that arise in a marine MM survey. First, we look at the effect of the relative vertical offset between the transmitter the receiver. We then show that it is necessary to acquire data over a large range of /H. Finally, we apply the derived formula to field data from the East Pacific ise. The transmitters are located 700 m below the sea surface, while the magnetometer is at 500 m depth. This simulates the case where the magnetometer is situated on the ridge axis without taking the bathymetry into account. Figure 8a shows the azimuthal B versus the normalized distance /H for the on-axis magnetometer. For comparison, the off-axis magnetic field is plotted also. Surprisingly, the amplitudes for the on-axis receiver are much larger about one order of magnitude than those for the off-axis receiver. This significant difference results purely from the vertical shift of receiver location. Without taking the transmitter-receiver geometric difference into account, the derived resistivities of the on-axis response varies from m, while the off-axis response yields values from m solid dots circles in Figure 8b, obtained using Wolfgram et al., Obviously, these results are unsatisfactory. When we use the new formula to obtain the apparent resistivity, both curves solid dashed lines in Figure 8b offer a good approximation to the model value 10.m. Effect of normalized range Analysis of the apparent-resistivity curve reveals the importance of the normalized distance on the data interpretation survey de- Effect of Tx-x vertical offset Although marine MM surveys have been conducted in several places Edwards et al., 1985; Evans et al., 1998; Evans et al., 00, we feel there are still some important practical questions to answer in order to apply this method more effectively. For simplicity of data processing interpretation, it usually is assumed that the transmitter receiver are located at the same depth below the sea surface. In practice, receivers are always dropped on the seafloor. However, the transmitter wire is often hanging in the seawater, not in contact with the seafloor, or the transmitter receiver are located at different depths because of the bathymetry of the seafloor. For example, the receiver might be on the ridge crest with the transmitter deployed at a deeper depth. If the vertical distance between the lower end of the transmitter wire the magnetometer is much less than the thickness of the seawater e.g., the ratio is 1%, the geometric difference may be negligible. However, if the ratio is approximately 10%, then the difference has a significant effect on the measured magnetic fields. Without taking the geometric difference into account, the interpretation will be compromised. A synthetic example is presented in Figure 8. Figure 8c assumes a layered model where the resistivities of the seawater seafloor are m, respectively. Figure 8. Effect of the on-axis off-axis magnetometers on a the observed magnetic fields b the derived apparent resistivity. c The 1D model used to investigate the effect.

8 G80 Chen Oldenburg sign. As shown in Figure 9a, the apparent-resistivity curve obtained with a normalized range 0.04 /H 4 labeled full Figure 9. Effect of the normalized range on data inversion survey design. a apparent resistivities for the three-layer seafloor model. b ecovered 1D resistivities with a small range /H = a full data range range clearly shows a three-layer model of the seafloor. The inset is the true 1D model. When we carry out a 1D inversion based upon a generalized cross-validation technique with full-range data, the recovered structure reveals the lower-resistivity layer in a three-layer model see Figure 9b. Conversely, if we only use the data in the normalized range 0.3 /H 4 labeled small range, the recovered model from a 1D inversion indicates a two-layer structure. While this can be explained easily from the apparent-resistivity curve, it is not as obvious if we look at the magnetic-field curve. This simple example suggests that if the normalized distance is not covered widely enough, we will likely miss the shallow-resistivity information, resulting in a poor 1D model. In this regard, choice of the normalized range has a definitive impact on the survey design. Figure 10. Apparent-resistivity map associated with instrument five at EP using a the Wolfgram et al formula b our formula. c Comparison of the apparent resistivity along the x 5-A profile marked in a b. The large red dot is the location of x instrument five. Note that the apparent resistivity computed for each transmitter is plotted at the transmitter location. The color scales in.m are slightly different, for it is difficult to make them identical.

9 Apparent resistivity for marine MM G81 Data set A marine MM experiment recently was conducted at the East Pacific ise EP to study the electrical resistivity of the shallow crust in the vicinity of the ridge Evans et al., 00. More than 00 transmitter bipoles 10 magnetometers were deployed in this experiment. The magnetometers could be categorized into two groups: on axis off axis. For the on axis magnetometers, the depth of the receiver was approximately 500 m; the depth for the transmitters varied from m. We chose instrument five, which was on the crest of the ridge, its associated 160 transmitters as an example. The apparent resistivities obtained using Wolfgram et al our formulas are shown in Figures 10a 10b. There are some similarities in these two plots. It appears that the shallow material has low resistivity, deep material has a relatively increasing resistivity. The difference can be seen from the apparent-resistivity profile along receiver 5-A see Figure 10c. Our curve shows a resistivity low between km, while Wolfgram et al does not. This resistivity low suggests there may be a fairly low-resistive layer at depth. Because of limited information, we cannot make the judgment that our result is better than that of Wolfgram et al for this example. More work on a 3D inversion must be carried out to obtain a 3D electrical structure in this region. CONCLUSION We have derived a new apparent-resistivity formula based upon the semianalytic expression for the magnetic field resulting from a semi-infinite electrode source buried in a 1D earth. The new formula is superior to the two most commonly used formulas in that it is accurate for a full range of the normalized transmitter-receiver distance. We have also investigated the effects of transmitter-receiver geometric difference the choice of normalized range on data interpretation survey design. The utility of the derived formula is demonstrated with synthetic field data sets. We believe that first-order approximate resistivity information can be obtained by converting the observed magnetic field to apparent resistivity, that this can assist data interpretation, survey design, estimation of a background model for 3D inversion. ACKNOWLEDGMENTS The work presented here was funded by NSEC the IMAGE Consortium, of which the following are members: AGIP, Anglo American Corporation, BHP Billiton, EMI Inc., Falconbridge Ltd., INCO Exploration Technical Services Inc., Kennecott Exploration, MIM Exploration Party Ltd., Muskox Minerals Corp., Newmont Exploration Ltd., Placer Dome Inc., Teck Cominco Ltd. We are grateful for their participation. Thanks also go to obert Evans of Woods Hole Oceanography Institute for providing the East Pacific ise data marine MM background information. The editing staff, including Jerry Schuster, Yonghe Sun, Mark Everett, three anonymous reviewers, offered insightful comments on the original manuscript, prompting a much-refined version. We are indebted for their help. EFEENCES Chave, A. D., S. C. Constable,. N. Edwards, 1991, Electrical exploration methods for the seafloor, in M. N. Nabighian, ed., Electromagnetic methods in applied geophysics: Investigations in geophysics: SEG, Chen, J., E. Haber, D. W. Oldenburg, 00, Three-dimensional numerical modeling inversion of magnetometric resistivity data: Geophysical Journal International, 149, Chen, J., D. W. Oldenburg, 004, Magnetic electric fields of direct currents in a layered earth: Exploration Geophysics 35, Edwards,. N., L. K. Law, P. A. Wolfgram, D. C. Nobes, M. N. Bone, D. F. Trigg, J. M. DeLaurier, 1985, First results of the MOSES experiment: Sea sediment conductivity thickness determination, Bute Inlet, British Columbia, by magnetometric offshore electrical sounding: Geophysics, 50, Edwards,. N., M. N. Nabighian, 1991, The magnetometric resistivity method, in M. N. Nabighian, ed., Electromagnetic methods in applied geophysics: Investigations in geophysics: SEG, Evans,. L., S. C. Webb, the IFT-UMC Team, 00, Crustal resistivity structure at 9 50N on the East Pacific ise: esults of an electromagnetic survey: Geophysical esearch Letters, 9, /001- GL Evans,. L., S. C. Webb, M. Jegen, K. Sananikone, 1998, Hydrothermal circulation at the cleft-vance overlapping spreading center: esults of a magnetometric resistivity survey: Journal of Geophysical esearch 103, Wolfgram, P. A.,. N. Edwards, L. K. Law, M. N. Bone, 1986, Polymetallic sulfide exploration on the deep sea floor: The feasibility of the MINI-MOSES technique: Geophysics 51,

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