Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2013) 194, Advance Access publication 2013 April 12 doi: /gi/ggt115 Practical incorporation of local and regional topography in three-dimensional inversion of deep ocean magnetotelluric data Kiyoshi Baba, 1 Noriko Tada, 2 Hisashi Utada 1 and Weerachai Siripunvaraporn 3,4 1 Earthquake Research Institute, University of Tokyo, 1-1-1, Yayoi, Bunkyo-ku, Tokyo , Japan. kbaba@eri.u-tokyo.ac.p 2 Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, 2-15, Natsushima, Yokosuka, Kanagawa , Japan 3 Faculty of Science, Mahidol University, Rama 6 Road, Rachatawee, Bangkok 10400, Thailand 4 ThEP, Commission of Higher Education, 328, Si Ayuthaya Road, Bangkok 10400, Thailand GJI Marine geosciences and applied geophysics Accepted 2013 March 20. Received 2013 February 25; in original form 2012 May 18 SUMMARY We propose a three-dimensional (3-D) inversion scheme for deep ocean magnetotelluric (MT) data that enables us to incorporate both the local small-scale topography effect and regional large-scale topography effect, whose length scale is comparable with that of the mantle structure to be resolved by the inversion. We assume that the MT impedance tensor Z is approximately equal to the MT response of the total structure Z ts, expressed by the product of the local topographic distortion term D lt and the response to the regional structure that consists of the regional topography over the mantle electrical conductivity structure Z Z ts = D lt. We also assume that D lt may be treated as a site correction term, which is not dependent on the conductivity structure to be solved in the inversion. D lt is calculated before the inversion iteration is carried out through the forward modelling of Z ts and using a good initial guess for the mantle structure. However, the initial Z ts is separately modelled by another forward modelling program, which is more efficient in terms of computation cost. The model space of the inversion includes only the regional structure so that the modelled response is.thus,in the inversion, Z ts is calculated from and D lt ; is modelled at every iteration but D lt is fixed throughout the inversion. The sensitivity of Z ts to the conductivity is calculated considering D lt as a constant. This scheme requires modelling only of the regional scale structure in the inversion process, saving computational resources and time, which is critical in the 3-D case. Tests using a synthetic model and data demonstrate that incorporating the local topography effect in this scheme was successful and produced an accurate reconstruction of the given mantle structure. Key words: Numerical approximations and analysis; Magnetotellurics; Geomagnetic induction; Marine electromagnetics. 1 INTRODUCTION Deep ocean magnetotelluric (MT) studies have been utilized to image electrical conductivity distributions of oceanic upper mantle, and they have contributed to the discussion of mantle dynamics in various tectonic fields (e.g. Constable & Heinson 2004; Baba et al. 2006; Matsuno et al. 2010). Considering topographic change and land ocean distribution has been one of the critical issues for marine MT studies, because the large contrast in the conductivity between seawater and crustal rocks severely distorts the MT response functions on the seafloor (e.g. Worsewski et al. 2010; Key & Constable 2011). This significant effect is produced not only by regional large-scale topography and land ocean distribution on a length scale comparable to that of the target mantle structure but also by local and much smaller-scale topography around observation sites. It is difficult for numerical modelling to simultaneously resolve the small-scale heterogeneity that produces distortion and also include the much larger scale heterogeneous structure of the underlying mantle, which is of geophysical interest. This is because the linear system of the forward problem to be solved becomes huge if the full scale topography is incorporated into an electrical conductivity structure model by fine numerical blocks. This problem is even more critical for inversion which performs many forward calculations. A typical example is a study introduced by Baba et al. (2010), in which deep ocean MT data were obtained as part of the Stagnant Slab Proect (Shiobara et al. 2009), hereafter referred to as SSP, for imaging the upper mantle conductivity structure beneath the Philippine Sea. The observation array covers an area of approximately km 2 in and around the Philippine Sea Plate, comprising of 25 deep ocean MT sites at km intervals. The maor distinct large-scale surface heterogeneities are the coast lines, maor trenches, troughs and ridges that characterize the basins. Their wavelength is more than several tens of kilometres and can be easily recognized by satellite-based global topography map data such as ETOPO2 given by the National Geophysical Data 348 C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society

2 Incorporating topography in 3-D MT inversion 349 Center (NGDC, last accessed 2013 April 8). Typical small-scale surface heterogeneities are the semilinear abyssal hills and valleys associated with past seafloor spreading, etc., which have the length scale of several kilometres or less. This type of local topographic change is observed by shipboard multinarrow beam sounding systems especially at sites in the western Shikoku Parece Vela Basin. For modelling and inversion analysis of such data sets, it is ineffective and impractical to make the numerical blocks finer only for the near surface heterogeneity. In this study, we introduce a practical scheme of threedimensional (3-D) inversion analysis of deep ocean MT data, which accounts for both the local and the regional topography effects. We model the local small-scale topography effect for each observation site separately and obtain the distortion term on the MT impedance tensor. The model space in the inversion consists of only the regional structures with a moderate number of numerical blocks, allowing us to perform the inversion analysis with practical computational resources. We extended our previous studies about 3-D forward/inversion based on finite difference methods (Baba & Seama 2002; Siripunvaraporn et al. 2005; Tada et al. 2012) to achieve the above scheme. The scheme can be applied for land MT data as well as for seafloor MT data because the methods can treat both land and seafloor topography by the same manner. In this paper, we suppose applications to seafloor data because the topographic effect is generally severe. There are some studies about 2-D modelling using unstructured finite element methods (e.g. Key & Weiss 2006; Franke et al. 2007; Li & Pek 2008) and their extension to 3-D inversion may be another possibility to deal with the problem although we do not discuss it further in this paper. In the following sections, we describe the details of our method and demonstrate some results of synthetic inversion tests. modelling are described in Section 2.1. In the inversion process, the model space is the same as that for the forward modelling of. The forward part of the inversion produces. D lt is applied to to obtain Z ts and the sensitivity of Z ts is calculated to update the conductivity model during the iterations. Details of this inversion process are described in Section Forward modelling to obtain Z ts To obtain the MT response to the total structure model Z ts for each site, we apply a 3-D forward modelling based on a finite difference method using staggered grids, FS3D (Baba & Seama 2002), which can treat precise seafloor topography relatively inexpensively. However, modelling full scale topography over a large area still requires a high computational cost even when using FS3D; thus, it is not practical for the modelling to apply FS3D with its original manner to a large 2-D array such as the SSP data. Therefore, we introduce a two-stage modelling scheme that simulates the effect of local small-scale topography around each observation site separately from the response to the regional large-scale topography (including the land ocean distribution) and subsurface heterogeneous structure. This scheme enables us to account for both the local small-scale topography and the regional large-scale topography relatively inexpensively. A schematic outline of the two-stage modelling is shown in Fig. 1. In the first stage, we conduct forward modelling to a large model that 2 METHOD In this study, first, we consider the effects of both the local smallscale topography and the regional structure by simulating the local small-scale topography effect as a site correction term on the MT impedance tensor for each observation site, and then apply it in the inversion process. We assume that the MT response Z(r, T )isapproximately equal to the MT response to the total structure Z ts (r, T ), expressed by multiplication of the local topographic distortion term D lt (r, T ) by the response to the regional structure (r, T )that consists of the regional topography over the mantle electrical conductivity structure, Z(r, T ) Z ts (r, T ) = D lt (r, T ) (r, T ), (1) where r is position and T is period. Z, Z ts, D lt and are all 2 2 complex tensors and hereafter their dependence on r and T will be implicit. Baba & Chave (2005) and Matsuno et al. (2007) introduced a similar relationship for the correction of the topographic effect on MT responses. The difference here is that the distortion term D lt includes only the local small-scale topography effect while the regional large-scale topography effect is included in.first,we calculate D lt by forward modelling of Z ts and by assuming subsurface structure. is simulated by a model expressed by moderate-size numerical blocks that are ust sufficient to resolve the regional structure of geophysical interest. However, Z ts is simulated by a total structure model that consists of both regional large-scale structure and local small-scale structure with finer meshes; thus, special treatment is introduced to achieve this. D lt is calculated from the relation (1) once Z ts and are obtained. Details of the forward Figure 1. Schematic diagram of the model spaces for the large and small models. (a) The large model (gray box) covers all observation sites (inverted cones) by moderate-dimension numerical blocks, which is used for the first stage modelling. The small models (black boxes) are prepared for each observation site with finer numerical blocks, which are used for the second stage modelling. (b) Horizontal grids of the large (gray lines) and small (black lines) models. The initial and boundary values of the magnetic field to be solved in the second stage modelling (arrow with square tail) is obtained by bi-linear interpolation of the four surrounding values (arrows with circle tail), which is the solution of the first stage modelling.

3 350 K. Baba et al. contains large-scale regional structure (without local small-scale topography) surrounding the whole observation array and which is discretized by a moderate-dimension numerical mesh. In the second stage, forward modelling to a smaller model space that includes local small-scale topography with a finer mesh is conducted for each observation site. FS3D solves the magnetic field values on the staggered grid by an iterative method and output them. The magnetic field values output in the first stage modelling are interpolated for the grids of the small model. In this study, we use identical grids in the vertical direction for the large and small models for simplicity, but the basal depth of the small model may be shallower than that of the large model. Each magnetic field component on the small model is calculated by bi-linear interpolation of four values on the large model at the same depth (Fig. 1b). Then, the obtained magnetic field values are input in the second stage forward modelling as the initial and boundary values of the equations to be solved. The MT impedance calculated from the electromagnetic field simulated by the second stage modelling will include the effects of both regional large-scale structure and local small-scale topography. This two-stage modelling was accurate enough to be compared with the brute-force one-stage modelling within the relative residuals of 1.3 per cent and 4.0 per cent for the off-diagonal and diagonal elements, respectively, as described in Appendix. The differences were comparable to or less than typical standard errors of real data. The advantages of the two-stage modelling are as follows: (1) lower computational cost and (2) easier grid design. As shown later Figure 2. Topography/bathymetry models for synthetic inversion tests. (a) Regional large-scale topography model, (b) local small-scale topography model for site T08 and (c) local small-scale topography model for site T13. Crosses are the site locations for which the synthetic data are produced. Green and red crosses indicate the location of sites T08 and T13, respectively. Purple dashed line in (a) is the area demonstrating the mantle structure models shown in Fig. 3. Black dotted lines in (b) and (c) indicate the horizontal mesh of the regional model.

4 Incorporating topography in 3-D MT inversion 351 for the case of the SSP array, we discretized a km 2 area by using equal-sized (50 50 km 2 ) grids to cover the array in the large model while the small model was expressed by unequal-sized grids (1 1km 2 at the centre) around each site. If we discretized the same area in the large model by 1 1km 2,we would require 2500 times the number of grids, which is impractical. Although one can use unequal-sized grids that are finer near each of observation sites, it is neither efficient to reduce the total number of grids nor easy to design the grids in a general 2-D array with irregularly located sites. For the two-stage modelling scheme, once the small model grids are designed so as to locate the site exactly at the centre of the modelling area, the same grid design may be applied to all sites. Therefore, the need for complicated grid design work or a superior grid generator is eliminated. 2.2 Inversion We assume a regularized inversion that minimizes an obective function consisting of a model misfit term and a data misfit term, (m) = (m m p ) T C 1 m (m m p) + λ 1 {d F(m)} T C 1 d {d F(m)}, (2) where d is the data parameter vector that contains the observed MT impedances Z, F(m) is the forward response to the model parameter vector m that contains the logarithmic resistivity or the inverse of the conductivity log ρ = log σ 1 of each model block, m p is the prior model vector, C m is a model covariance matrix, C d is a data covariance matrix and λ is a parameter that balances the data misfit and model misfit terms. The model space of the inversion expresses only the regional structure; thus, the forward response F(m) isa vector consisting of the components of Z ts = in a conventional inversion scheme. In this study, we assess Z ts for structure including local smallscale topography by (1). Then, we first calculate the local topographic distortion term D lt from 0 and Zts 0,whereZrs 0 and Zts 0 are the forward responses for an initial model m 0 calculated by the inversion program and by the separate forward modelling described in the previous section, respectively. We assume that the effect of the local small-scale topography is not significantly coupled with the mantle heterogeneous structure and fix D lt during the inversion iterations. Consequently, the data misfit term at the kth iteration is calculated for Z and Z ts k = Dlt k and the sensitivity of Zts k to m k is calculated as, Z ts m Z ts xy m Z ts yx m Z ts m = Dlt m + Dlt m + Dlt xy yx m + Dlt xy yx m D lt + D lt yx xy, (3) m m = Dlt xy m + Dlt xy m + Dlt xy m + Dlt xy m Z D lt xy rs + D lt xy, (4) m m = Dlt yx m + Dlt yx m + Dlt yx m + Dlt yx m D lt yx + D lt yx, (5) m m = Dlt yx xy m + Dlt yx xy m + Dlt m + Dlt m Z D lt xy rs yx + D lt, (6) m m Figure 3. 3-D Mantle structure model for the synthetic inversion tests. Top three panels show the horizontal slices at depths of 46, 90 and 124 km. Bottom three panels show the y z sections at x = 275, x = 0andx = 275 km. Gray dashed lines indicate the position of the horizontal slices or vertical sections. Crosses are the site locations. The background 1-D model is taken from the representative model for the Philippine Sea mantle (Baba et al. 2010). The conductivity values are given by the colour scale shown at the bottom, and those of the conductive and resistive bodies are 0.1 and S m 1, respectively.

5 352 K. Baba et al. Figure 4. MT responses in terms of apparent resistivity and phase for sites T08 and T13. The four elements of the MT impedance tensor are plotted in different colours. Symbols with error bars are the synthetic responses. Dotted lines are the first stage responses of the two-stage forward modelling for producing the synthetic responses. Solid and dashed lines are the model responses for inversion A (new method) and for inversion B (conventional method) described in Section 3. where is an index for the conductivity blocks and k is omitted for simplicity. The sensitivity of in the above equations may be calculated by a conventional method (e.g. Newman & Alumbaugh 2000). The assumption that D lt is independent of the conductivity structure (model parameter) is expected to hold if a good initial guess to the model parameter is given, as discussed in Section 4. We incorporate the scheme mentioned earlier in a 3-D MT inversion program given by Tada et al. (2012), which is based on the inversion developed by Siripunvaraporn et al. (2005) with expansions to incorporate the topography in the model and to calculate the MT response on the seafloor. Consequently, the actual inversion has a second loop to seek the best λ at each iteration that minimizes (2) for different λ, according to the original inversion algorithm. For further details regarding the inversion algorithm, see Siripunvaraporn et al. (2005). The inversion program was modified for three calculations: (i) D lt from 0 sensitivity of Z ts given in (3 6). and Zts 0, (ii) Zts by (1) and (iii) the 3 SYNTHETIC TESTS We tested the inversion scheme by using a synthetic model and data. The synthetic model consists of a 3-D surface heterogeneity (topography and bathymetry) and a 3-D mantle structure. The surface heterogeneity model was produced from real topography data in and around the Philippine Sea by averaging the elevation or depth within given horizontal meshes. The regional large-scale topographic model was produced from 2 min-mesh data, ETOPO2 (Fig. 2a). The central km 2 area was discretized every 50 km and the outer area was discretized more coarsely as the distance from the centre increased. The entire dimension of the large model covered a horizontal area of km 2, with a basal depth of 1026 km below the sea surface. The local small-scale topography model was produced by a combination of ETOPO2 and finer 250 m-mesh data based on multi-narrow beam soundings, which were collected by research cruises of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC). The central km 2 area was discretized every 1 km and the outer area was discretized with a grid that became coarser (up to 50 km) further from the centre. The observation sites were located at the centre of each modelling area. The entire dimension of the small model covered a horizontal area of km 2 and a basal depth of 460 km. The mesh design was identical for all the sites. Examples of the local topography model for sites T08 and T13 in the western Shikoku Basin (green and red crosses,

6 Incorporating topography in 3-D MT inversion 353 respectively, in Fig. 2a) are shown in Figs 2(b) and (c), respectively. The bathymetry around site T08 is relatively rougher than that around T13; thus, suggesting a different effect of the local topography. The conductivity for the continental crust and seawater was set to 0.01 and 3.2 S m 1, respectively. The mantle structure model (Fig. 3) consists of conductive (0.1 S m 1 ) and resistive (0.001 S m 1 ) bodies superimposed on a background 1-D layered structure, which is the representative structure for the Philippine Sea mantle obtained by Baba et al. (2010). The synthetic MT impedances are produced by two-stage forward modelling for 25 sites, as shown in Fig. 2(a). We added 1.5 per cent Gaussian noise to the MT impedances, which is equivalent to 3.0 per cent for the apparent resistivity and 0.86 for the impedance phase. The error of the synthetic MT impedance is set as 1.5 per cent of the absolute value. The number of sites and their locations correspond to those of the real observation sites reported by Baba et al. (2010), because dense multi-narrow beam sounding data are available for the area around those sites. Sixteen periods from 320 to s were calculated for the synthetic data, a similar range to that covered by the real data. Fig. 4 shows examples of the synthetic MT responses for sites T08 and T13. The general features of the responses of these sites are similar to each other because the both sites locate in the southern part of the Shikoku Basin. However, significant differences are recognized in the diagonal elements especially for the periods shorter than s. The diagonal apparent resistivities for site T08 increase with decreasing period, while those for site T13 do not experience significant change. The corresponding phases are also very different in the shorter period range between the two sites. The feature of the diagonal responses for site T08 is mainly attributed to the rougher local small-scale topography around this site. Dotted lines in Fig. 4 shows the first stage responses of the two-stage forward modelling for producing the synthethic responses, which only regional large-scale topographic effect is taken into account. The difference between the synthetic and first-stage responses for T08 are larger for the diagonal elements in shorter period range than those for T13, verifing the stronger effect of the local small-scale topography at around T08. Two inversions, A and B, were introduced to demonstrate the performance of the proposed inversion scheme and highlight that it is essential to take the small-scale topography into account. For inversion A, we applied the new scheme described in the previous section. For inversion B, we simply inverted the synthetic data using the conventional inversion scheme of Tada et al. (2012), in which the effect of the local small-scale topography is not taken into account, even though the synthetic data are affected by the topography. For both cases, we applied error floors of 2.5 and 5.0 per cent for the offdiagonal and diagonal elements of the MT impedance, respectively, and the synthetic background 1-D model as the prior and initial models. The maximum number of inversion iterations was 10, and we regarded the model, for which the root mean square (rms) misfit of the data (MT responses) was smallest, as the best model. The rms of the data misfit, α, isdefinedas α = 1 N d [ di F i (m) N d ε i i i ] 2 = 1 N d ( Zi Z ts ) 2 i, (7) N d ε i Figure 5. Variation of the rms of the data misfit, α, with inversion iterations. Diamonds and circles correspond to inversion A and B, respectively. Double symbols indicate the minimum values. where N d is the total number of data and ε i is the error of Z i.the minimum αs were obtained at the third and tenth iterations for inversion A and B, respectively (Fig. 5). The best model of inversion A recovered well both the conductive and resistive anomalies in the mantle (Fig. 6a). Conversely, the best model of inversion B recovered the anomalies only weakly especially for the resistive anomaly (Fig. 6b). We plotted the residual of the inversion models to the synthetic model (referred to as model recovery ) in Fig. 7 to demonstrate the above features more clearly. The superior recovery of inversion A can be confirmed quantitatively by the rms of the model recovery, β,definedas β = 1 V total N m ( m inv m syn ) 2 V [ ( )] = 1 N m ρ inv 2 log V total ρ syn V, (8) where m inv = log ρ inv, m syn = log ρ syn and V are the model parameters (logarithmic resistivity) recovered by the inversions, those for the synthetic structure, and the volume of the th block, respectively, N m is the number of model blocks to be assessed and V total is the total volume of N m blocks. β values are calculated for some spaces and listed in Table 1. The βs within the space 1275 km < = x < = 1275 km, 1275 km < = y < = 1275 km and km < = z < = 410 km, which is almost the same space as plotted in Figs 3, 6 and 7 (referred to as the plot space hereafter) are for inversion A and for inversion B. The βs for the conductive body for inversion A and B are and 0.384, while those for the resistive body are and 1.149, respectively, confirming that the MT data are relatively more sensitive to conductive bodies. For the result of inversion B, there is a false conductive anomaly at around x = 100 km, y = 300 km, z = 90 km even though the location is ust underneath the data site, where there should be a resistive anomaly (Fig. 6b, upper centre panel).

7 354 K. Baba et al. Figure 6. The best conductivity models recovered by inversions A (a) and B (b). Symbols and lines are the same as those in Fig. 3 but dotted lines indicate the locations of the resistive and conductive bodies in the synthetic model. The best α s for the two models were 1.71 and 9.41, respectively. The difference between the two α s and its significance compared to the input noises are obvious in the plot of the responses (Fig. 4). The data misfit for inversion B was worse than for inversion A, which is probably because of the effect of disregarding the local small-scale topography on both the forward and inverse problems. The effect on the forward problem alone can be seen at the zeroth iteration of the inversions because the forward responses are calculated for the

8 Incorporating topography in 3-D MT inversion 355 Figure 7. Model recovery for inversions A (a) and B (b). Symbols and lines are the same as those in Fig. 6. initial model, which are identical for both inversions. In the later iterations, the effect of the poorer model recovery mentioned earlier is superimposed on the forward effect in this case. These results suggest that taking into account the effect of the local small-scale topography significantly reduces the data misfit and contributes to a more reliable model. 4 MUTUAL COUPLING TESTS In this section, we present more practical applications of the new scheme and discuss the importance of a good initial guess of the subsurface structure. For the synthetic inversion tests demonstrated in the previous section, we used a synthetic background 1-D

9 356 K. Baba et al. Table 1. The rms s of the model recovery, β, for the synthetic inversion tests. A A B C D E Background (BG) Conductive body (CB) Resistive body (RB) Plot space (BG+CB+RB) structure as the prior and initial models. However, in practical cases, the background structure is unknown, and thus, it has to be estimated from observed data and/or independent information. In general, the use of a good initial model is the key in inversions of non-linear problems to obtain a reliable model with faster convergence. It is also desirable for the determination of the local topographic distortion term D lt in our inversion scheme, which is essentially dependent on the subsurface structure. However, in practice, D lt is calculated from the total and regional responses to the initial model (Z ts 0 and 0 ) and is fixed throughout the inversion iterations, based on the assumption that D lt does not change significantly with changes in subsurface structure. Thus, we assume that the mutual coupling between the effect of the local small-scale topography and the subsurface 3-D heterogeneities are negligibly weak. We now carry out two additional synthetic inversions (C and D) to discuss the initial model and the mutual coupling problems. We applied two different 1-D models for the prior/initial model of inversions C and D (Fig. 8). These models were estimated from the synthetic data Z, based on the procedure introduced by Baba et al. (2010), as follows. The square root of the determinant of the MT impedance tensor Z det was calculated for all sites, and the mean of Z det was inverted using Occam s inversion (Constable et al. 1987) to obtain the 1-D model. Further, to reduce the influence of Figure 8. Background 1-D conductivity profiles for the synthetic model and for the prior/initial models of inversions C and D. Dashed lines indicate the conductive and resistive bodies in the synthetic 3-D model (Fig. 3). the topographic effect on the 1-D model estimation, the following procedures were carried out. The topographic distortion D, which contains both regional large-scale and local small-scale effects in this case, was calculated by D = Z ts Z 1 1D,whereZts is the response to the total structure consisting of the topography over the 1-D model obtained above that can be simulated by the two-stage modelling scheme, and Z 1D is the response to the 1-D model. Next, Z of each site was corrected for the topographic effect by Z cor = D 1 Z.The 1-D model was estimated again in the same manner as above but using the corrected responses Z cor. We applied these 1-D models estimated from Z cor and from Z to the 3-D inversions C and D, respectively. The 1-D model for inversion C shows a better fit to the synthetic background 1-D structure although a slight difference can be observed at depths of km, where the conductive and resistive bodies are located in the synthetic 3-D model. However, the 1-D model for inversion D is significantly biased towards the conductive side below approximately 70 km because of the topographic effect. Comparison of the results for inversions A, C and D suggests the influence of prior/inital model in the 3-D inversions. Inversion C recovered a 3-D structure closer to the true model than inversion D, although neither is as good as inversion A that used the true background structure as prior/inital model. Figs 9 and 10 show the results of inversions C and D. For inversion C, the βs forthe conductive and resistive bodies are and 0.812, respectively, which is similar to the level of inversion A. Conversely, the β for the background 1-D structure is 0.202, which is worse than that of inversion A (Table 1), causing a poorer β of for the whole plot space. Compared with the synthetic value, the background structure of the inverted model is more resistive at km depth and slightly more conductive at approximately 120 km depth, especially for the regions further away from the sites (Figs 9a and 10a). For inversion D, the trend of the model recovery is qualitatively similar to that of inversion C but the recovery is quantitatively poorer (Figs 9b and 10b). The only exception is the β for the conductive body, which is the best among the four inversions (Table 1). This result is mainly because the prior/initial model of inversion D happened to have conductivity values similar to those of the conductive body around 120 km depth (Fig. 8); hence, this does not imply that the body is fully resolved by the data. In fact, the recovered conductivity is closer to the true value in the regions away from the horizontal location of the data sites (Figs 9b and 10b, upper right panels). The dependence of D lt on the subsurface structure is examined by comparing D lt s for the synthetic and inverted models. The synthetic D lt for each site can be calculated using (1), where Z ts and are obtained by the forward modelling to the synthetic 3-D structure. Fig. 11 shows examples of D lt for sites T08 and T13. Without the effect of the local small-scale topography, D lt is a unit matrix by definition. Thus, any diversity from the unit matrix would indicate the effect of the local small-scale topography. For sites T08 and T13, each component of D lt is further diverged from the value of the unit matrix as the period decreases (especially < 10 4 s); the real parts are significantly different from one or zero and the imaginary parts are non-zero. For the longer periods, the imaginary parts approach zero, but the real parts of the diagonal components are still significantly different from one (e.g. D lt for T08 and D lt for T13). These facts suggest that the effect of the local small-scale topography of the synthetic model is not galvanic but rather inductive for short periods but approaches galvanic as the period increases. The broad features of the inverted D lt are similar to those of the synthetic D lt for all of the inversions although some

10 Incorporating topography in 3-D MT inversion 357 Figure 9. The conductivity models recovered by inversions C (a) and D (b). Symbols and lines are the same as those in Fig. 6.

11 358 K. Baba et al. Figure 10. Model recovery for inversions C (a) and D (b). Symbols and lines are the same as those in Fig. 6.

12 Incorporating topography in 3-D MT inversion 359 Figure 11. D lt for site T08 and T13 applied to inversions A (red symbols), C (green symbols) and D (blue symbols). Cross and plus symbols denote the real and imaginary parts, respectively. Gray solid and dashed lines are those for the synthetic D lt. Table 2. The rms s of the distortion recovery, γ, for inversions A, A, C and D and for the case of unit matrix D lt. A A C D Unit Matrix γ discrepancies can be seen. To assess the discrepancies for all the sites quantitatively, we introduce rms of distortion recovery, γ, defined as follows: γ = 1 N d N d i ( D ltinv i ) 2. D ltsyn i (9) The γ s for inversions A, C and D and the γ for a virtual case of D lt as a unit matrix for all sites are listed in Table 2. The γ for the virtual case indicates the strength of the effect of the local small-scale topography of the synthetic model. The former three γ s are close to zero and much smaller than the γ for the virtual case; thus, confirming that the effect is significant and cannot be ignored and that D lt in the three inversions broadly fits the synthetic D lt. Furthermore, γ is the smallest for inversion A and is worst for inversion D, which correlates with the diversity of the initial model of each inversion from the synthetic subsurface structure. This result confirms that D lt is dependent on the subsurface structure and a better subsurface structure model improves the estimate of D lt. The dependency of D lt on the initial model is not a significant problem for model recovery, at least in inversions A and C. Distinct false structure as seen in inversion B is not observed in the models recovered by the other three inversions. The difference of the β seems to be largely controlled by the prior model. To demonstrate it, further tests were conducted using inversions E and A. Inversion E was run with the same condition as inversion C but the synthetic background 1-D model was given as the prior model. Thus, D lt in inversions C and E is identical. The impact of the prior model alone can be seen through comparison of the two inversions. The resultant 3-D model of inversion E is closer to the synthetic model (not shown in a figure) with β for the plot space of (Table 1). Inversion A used the 3-D model recovered by inversion A as the prior and initial models. The model recovery for inversion A improved, as seen by the value of the β s for the conductive/resistive bodies and the plot space (Table 1). However, γ for inversion A was rather worse than that for inversion A and similar to that of inversion C (Table 2). This feature is more clearly seen in Fig. 12, which indicates the distribution of the residuals between D lt obtained by the inversions and the synthetic D lt for all sites. We conclude from these observations that the accuracy of the estimation of D lt is not so critical if the initial model approximates the true structure to some extent as in the case of inversion C. Moreover, the assumption that D lt is constant during the inversion iterations is practically valid. The above results also confirm the importance of a good initial

13 360 K. Baba et al. REFERENCES Figure 12. Absolute residuals between each element of D lt and the synthetic D lt for all sites, obtained by inversions A, A and C. guess for the deep ocean MT inversions considered here. This study demonstrated that the estimation of 1-D structure, as in inversion C, is one of the best ways to determine the initial model and D lt. Selection of a prior model is a general issue for any inversions based on (2), and thus, out of the scope of this study. We note that updating the prior and initial models using an inversion result, as in inversion A, can be useful to validate the mutual coupling of D lt and to estimate a more reliable structure. These issues should be examined in each application to real data. 5 CONCLUSIONS We proposed a new inversion scheme that considers the effects of both regional large-scale and local small-scale topography on deep ocean MT data. The effect of the local small-scale topography was first modelled separately and applied as a site correction term in the inversion process, assuming the effect is not significantly coupled with the mantle structure of geophysical interest. This assumption is practically valid if a good initial guess of the subsurface structure is given. The scheme was tested using a synthetic model and data. The results showed that when the local small-scale topography effect is significant, the newly proposed scheme is essential to recover the reliable structures. The conventional scheme produced a false structure. These results indicate the importance of dealing with the local small-scale topography effect in deep ocean MT studies. ACKNOWLEDGEMENTS The authors thank the editor (Oliver Ritter), Rob L. Evans and two anonymous referees for their helpful comments which improved the manuscript, and Enago ( last accessed 2013 April 8) for the English language review. The bathymetry data based on multinarrow beam soundings used in this study were provided by the JAMSTEC Data Site for Research Cruises. All figures were produced using GMT software (Wessel & Smith 1998). This work was supported by Grants-in-Aid for Scientific Research , the Japan Society for the Promotion of Science (JSPS) and Thailand Research Fund (TRF:RMU ). Baba, K. & Seama, N., A new technique for the incorporation of seafloor topography in electromagnetic modelling, Geophys. J. Int., 150, Baba, K. & Chave, A.D., Correction of seafloor magnetotelluric data for topographic effects during inversion, J. geophys. Res., 110, B12105, doi: /2004jb Baba, K., Chave, A.D., Evans, R.L., Hirth, G. & Mackie, R.L., Mantle dynamics beneath the East Pacific Rise at 17 S: Insights from the Mantle Electromagnetic and Tomography (MELT) experiment, J. geophys. 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Un., 90(9), Siripunvaraporn, W., Egbert, G., Lenbury, Y. & Uyeshima, M., Threedimensional magnetotelluric inversion: data-space method, Phys. Earth planet. Inter., 150, Tada, N., Baba, K., Siripunvaraporn, W., Uyeshima, M. & Utada, H., Approximate treatment of seafloor topographic effects in threedimensional marine magnetotelluric inversion, Earth Planets Space, 64, Wessel, P. & Smith, W.H.F., New, improved version of the generic mapping tools released, EOS, Trans. Am. geophys. Un., 79(47), 579. Worzewski, T., Jegen, M., Kopp, H., Brasse, H. & Castillo, W.T., Magnetotelluric image of the fluid cycle in the Costa Rican subduction zone, Nat. Geosci., 3, APPENDIX: ACCURACY OF TWO-STAGE MODELLING The accuracy of the two-stage forward modelling is tested by comparing the MT response with that calculated by the conventional method incorporating both regional large-scale and local smallscale topography directly. The synthetic model consists of real topography in the Philippine Sea (Fig. 2) and 0.01 S m 1 half-space beneath the seafloor. Only the local small-scale topography around site T08 is taken into account so that brute force modelling using

14 Incorporating topography in 3-D MT inversion 361 Figure A1. MT responses calculated by conventional modelling (gray lines) and the two-stage modelling (coloured symbols) for the accuracy test. Diamonds, pluses and crosses indicate the response to the models with basal depths of 150, 462 and 1028 km, respectively. Red, green and blue symbols indicate the responses to the models with horizontal dimensions of 250, 350 and 450 km, respectively. the conventional method would be possible with practical computational cost by setting a finer horizontal mesh only around the site. We carried out the two-stage modelling nine times, changing the dimension of the small model (250, 350 and 450 km in lateral dimension; 150, 462 and 1028 km for the basal depth) for the second stage to check the effect of the Dirichlet-type boundary condition of the model edges. The maximum basal depth was the same as that of the large model. The resultant MT responses in terms of apparent resistivity and phase are shown in Fig. A1. The responses calculated by the twostage method indicate a good reconstruction of the overall features of the responses calculated by conventional brute force modelling, especially for the off-diagonal elements. The difference in the response among the small models with various dimensions is small; it is only visible in the and elements around a few thousand seconds. Table A1 is the rms misfit of the relative residuals to the response by conventional modelling. The rms misfit is larger for the diagonal elements than for the off-diagonal elements. The max- Table A1. rms misfit of the relative residuals (per cent) for the MT responses calculated for the two-stage forward modelling. H.D. and B.D. indicate the horizontal dimension and the basal depth of the small model, respectively. H.D. for Z xy, yx (km) H.D. for Z, (km) B.D. (km) imum values are 1.32 and 3.96, respectively. It may be expected for a trend of the rms misfit that modelling a larger model would result in a smaller rms misfit if the boundary condition at the model edge affects the field calculated at the centre. However, we cannot see such a clear trend, suggesting that the effect of the boundary is comparable to or less than the accuracy of the numerical solver.