Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media

Transcription

1 Geophys. J. Int. (8) 75, doi:./j x x Adaptive finite element modelling of two-dimensional magnetotelluric fields in general anisotropic media Yuguo Li and Josef Pek Scripps Institution of Oceanography, University of California San Diego, 95 Gilman Drive, La Jolla, CA. Geophysical Institute, Acad. Sci. Czech Rep., v.v.i. Bocni II/4, CZ-43 Prague 4, Czech Republic Accepted 8 August. Received 8 August ; in original form 7 September GJI Marine geoscience INTRODUCTION Electrical anisotropy is recognized as an important factor, in attempts to understand magnetotelluric (MT) observations. To give a few examples, the Mantle Electromagnetic and Tomography (MELT) experiment (Evans et al. 999, 5) in the fast spreading southern East Pacific Rise revealed the pronounced electrical anisotropy in the lower crust and the upper mantle. The MT measurement from the New Zealand Southern Alps suggests that the average conductivity of the lower crust conductive zone seems to be much higher than that across the strike and may present anisotropy (Wannamaker et al. ). At the South Chilean continental margin, all real induction vectors, at long periods of approximately 3 s, point systematically NE for all sites in the survey area, and this behaviour can be explained by -D models with an anisotropic layer in the continental crust (Brasse et al. 5). For a common modelling practice, Pek & Verner (997) presented a finite difference algorithm for MT responses over -D anisotropic media. This algorithm has been extended to include topography and bathymetry (Pek & Toh ), where the topography and bathymetry are approximated by simple staircase functions. A finite element code for modelling MT fields in -D anisotropic structures was published by Li (). Recently, 3-D direct modelling SUMMARY An adaptive unstructured mesh finite element (FE) procedure is presented for improving the quality of numerical solutions to the magnetotelluric forward problem in a general -D anisotropic conductivity structure. We implement a self-adaptive, goal-oriented grid refinement algorithm in which a finite element analysis is performed on a sequence of refined meshes. The mesh refinement process is guided by a dual error estimate weighting to bias refinement towards elements that affect the solution at the EM receiver locations. We validate the finite element code against a layered -D model with a sea water layer. Further, we compare the FE results with those obtained by a finite-difference (FD) scheme for both a block seamountain and a sea bottom hill model. Both FE and FD schemes show very good agreement for the block seamountain model. For the sea bottom hill model, however, only on the flat seafloor segments both the FE and FD solutions fit very well, but on the seafloor slope, FD results are oscillating due to a simplistic staircase approximation of the bathymetric undulations. The FD scheme for -D anisotropic conductors, developed primarily for the modelling of magnetotelluric data on a flat Earth surface, is thus not an adequate tool for dealing with structures with sloping bathymetry and topography, whereas the FE method with adapting mesh can easily handle such structures at almost any level of complexity. Key words: Numerical approximation and analysis; Magnetotelluric. codes for anisotropic conductors have been presented, for example, by Weidelt (999), Wang & Fang (), Weiss & Newman () and Hou et al. (6). Modelling of sea bottom MT data puts special demands on the numerical algorithms. The seafloor MT fields from the MELT experiment are strongly distorted by the rugged mid-ocean ridge rise (Baba et al. 6). The topography presents numerical difficulty, because one must simultaneously resolve both the smallscale bathymetry that produces distortion and the much larger-scale conductivity structures of underlying mantle (Baba et al. 6). Moreover, the huge conductivity contrast between the highly conductive sea water and the resistive underlying seafloor can also cause numerical difficulties. These difficulties may be overcome when an unstructured mesh is employed (Franke et al. 7), which can accommodate small-scale structures using small elements and large-scale structures using large elements in the same grid. In this paper, we revisit the -D MT forward problem in a general anisotropic structure and present a new FE procedure. Our new code implements a fully unstructured triangular mesh, which readily allows us to model complex structures such as topography, bathymetry, dipping layers and so on. We have also incorporated a self-adaptive grid refinement technique, based on the dual error estimate weighting (Ovall 6), in which the global influence on 94 C 8 The Authors

2 FE modelling of -D MT fields in general anisotropic media 943 the local error is taken into consideration. This method has been recently shown to be effective for the -D MT (Key & Weiss 6) and marine controlled-source electro-magnetic (CSEM) forward problems (Li & Key 7). The paper is organized as follows. We briefly describe the finite element approximation of the MT forward problem in -D anisotropic Earth media. We then present the adaptive grid refinement scheme and validate the code against the -D model with a sea water layer. In the subsequent section, we discuss two simple marine MT examples and compare the performance of both the FE algorithm presented in this paper and the previously developed FD scheme (Pek & Toh ). Finally, we present a model of a hypothetic onshore volcano with rough topography undulation and illustrate the capability of the presented FE code for modelling complex conductivity structures. FINITE ELEMENT APPROXIMATION The finite element (FE) approximation of -D MT fields, in generally anisotropic conductivity structures is only summarized here, the reader is referred to Li () for all the details. We consider a -D model with an invariant conductivity in the structural strike direction parallel to the x-axis of a Cartesian system of coordinates, with the positive z-axis pointing downwards. Assuming a time variation e iωt, the governing equations for electromagnetic field in the quasi-stationary approximation are E = iωμ H, () H = σ E, () where σ σ σ xz σ = σ σ σ yz (3) σ zx σ zy σ zz is the electric conductivity tensor. In the Earth, this tensor is symmetric and positive definite, thus characterized by only six independent components. The tensor can be rotated into its principal axes and then can be described by its three principal values σ x, σ y and σ z and the corresponding three Euler angles α s (strike), α d (dipping) and α l (slant). For the -D conductivity model, eqs () and () can be combined to yield two coupled second-order partial differential equations for the strike-parallel components E x and H x : E x + CE x + A H x iωμ y B H x z (τ H x ) iωμ H x (AE x) y where + (BE x) z =, (4) =, (5) D = σ σ zz σ yz σ zy, A = (σ σ zy σ zx σ )/D, B = (σ zx σ yz σ σ zz )/D, C = σ + σ B + σ xz A, ( ) σ σ τ = yz. D σ zy σ zz To solve for the unknown fields E x and H x, inhomogeneous boundary conditions are applied on the outer boundaries of the model, constructed from -D solutions for the corresponding layered media at the left- and right-hand side of the -D model. Classical solutions of boundary value problems of partial differential equations may not exist, even for smooth data (Han 5). The development of the theory of Sobolev spaces and weak formulations eliminates this problem and provides a general framework to derive numerical methods (Han 5). The weak formulation of the boundary value problem of the partial differential eqs (4) and (5) is derived as follows. First, multiply eqs (4) and (5) by an arbitrary function δe x and δh x, respectively. Next, integrate the resulting equations over a model realm and perform integration calculations by using Green s formula and substitute the boundary conditions (Li ), yielding E x δe x d CE x δe x d iωμ + ( A H x y + B H x z ) δe x d =, δh x (τ H x )d + iωμ H x δh x d ( + A δh x + B δh ) (7) x E x d =. y z Using the inner product notation, eqs (6) and (7) can be expressed as B(u,v) =, (8) For eq. (6): B(u,v) = v (α u)d βuvd ( + A y q + B z q ) vd, u = E x, v = δe x, q = H x, α =, iωμ β = C. () For eq. (7): B(u,v) = v (α u)d + βuvd ( + A y v + B z v ) () qd u = H x, v = δh x, q = E x, α = τ, β = iωμ. () The model area is subdivided into triangular elements. The integrals in eqs (6) and (7) thus decompose into integrals for each triangle element. We assume that in each triangular element, both the electric and magnetic fields (E x and H x ) vary linearly and can be approximated by linear interpolation functions. Then the area integrals in eqs (6) and (7) over an element can be analytically evaluated. The integrals over all triangular elements can be assembled together to form the global FE system of equations Ku =, (3) where u is the column vector of order of n d (n d is the total number of vertices in ), consisting of the strike-parallel components E x and H x at all vertices. The global matrix K is sparse, symmetric and singular. In other words, the equation system (3) has no unique solution without the incorporation of appropriate boundary conditions. To obtain a unique solution, the electric and magnetic field values on the outer boundary must be incorporated in eq (3). Note that the symmetry of the global matrix must be preserved in the implementation of the Dirichlet boundary conditions. We adopted (6) (9) Journal compilation C 8 RAS

3 944 Y. Li and J. Pek the following approach. The off-diagonal entries of each row corresponding to a Dirichlet boundary node in the global matrix and the diagonal entry of that row, respectively, are replaced by s and, and the entry in the row corresponding to that boundary node in the zero column matrix of the right-hand side of eq. (3) is assigned the given value on that boundary node. To preserve the symmetry of the global matrix, the off-diagonal non-zero entries in the column corresponding to that boundary node in K are also replaced by zeros. This requires that the entries of the rows corresponding to the internal vertices in the zero column matrix must be modified and they are replaced by the negative boundary value on that boundary node times the elements of the corresponding column vector of K. This approach has the advantages of not changing the number of equations and hence simplifies the implementation of the Dirichlet boundary condition. The disadvantage of the procedure is that the system of equations contains some trivial equations corresponding to the outer boundary vertices. Once the strike-parallel field components E x and H x are obtained, the other two electric components E y and E z as well as two magnetic components H y and H z canbedeterminedbyspatially differentiating: E y = σ yz D E z = σ D H x y H x y + σ zz H x + BE x, (4) D z σ zy D H x z + AE x, (5) H y = iωμ E x z, (6) H z = E x iωμ y. (7) In a MT survey, receivers are placed on the air Earth surface for land MT or on the seafloor for marine MT and hence follow the local slope. If a receiver lies on a flat surface, it measures the horizontal and vertical components of the electric and magnetic fields. If a receiver lies on a local slope, it measures the electric and magnetic components along the slope (E and H ) and perpendicular to the slope (E and H ). The corresponding horizontal and vertical components can be derived by E y = E cos φ E sin φ, E z = E sin φ + E cos φ, (8) H y = H cos φ H z sin φ, H z = H sin φ + H cos φ, (9) where φ is a slope angle with respect to the horizontal axis y. 3 A POSTERIORI ERROR ESTIMATION AND ADAPTIVE GRID REFINEMENT The weak eqs (6) and (7) can be discretized and solved on a pregenerated, fixed finite element grid. However, the quality of the resulting numerical solution depends largely on the underlying mesh. The design of a reasonable mesh may present difficulty. Adaptive finite element methods can automatically improve mesh design and offer reliable solution at a reasonable computational cost. This algorithm starts with a coarse mesh, and successively refined meshes are generated according to a posteriori error estimator, which assesses the accuracy of the FE solution. A variety of a posteriori error estimates has been proposed last decades, see the monographs by Ainsworth & Oden () and Han (5) for overview of this field. In this paper, we use the dual error estimate weighting (Oval 4, 6) in which the global influence on the local error is taken into consideration. This method has been recently shown to be effective for the -D MT (Key & Weiss 6) and marine CSEM forward problems (Li & Key 7). After the solution on a coarse mesh is obtained, the error for each element is calculated by ˆη e = η e η e, () with η e = α (R I ) u h L (e), η e = (R I ) w h L (e), () where I is the identity operator and u h is the gradient of a FE solution u h. R is a gradient recovery operator and w h is the FE solution gradient of the dual problem. The details about the derivation of eq. () are given in Appendix A. In our implementation, we refine 5 to per cent of elements with the largest ηˆ e values. The refinement of triangles is carried out by calling Triangle (Shewchuk 997, ), an automatic -D Delaunay generator. The areas of the triangular elements selected for refinement are reduced by at least 5 per cent of their current sizes, and the inner angles of the elements are required to be larger than 5 to avoid generating sliver shaped elements, which could degrade the accuracy of the FE solution. After an improved mesh is generated, the problem is solved again on the new mesh and the error indicator ηˆ e is updated. This process is repeated until the required level of accuracy is achieved. The convergence of the FE solution is measured by computing the rms and the relative difference of the off-diagonal apparent resistivities and phases on the current and previous mesh at the receiver locations. The relative difference δ p m,s i in both the apparent resistivity and the phase p m,s i (i =,, 4) for refinement mesh m and site s is calculated by δp m,s i = pm,s i p m,s i p m,s i () and the rms is defined as rms = 4 ns ( ) δp m,s, i (3) n i= s= where n = 4 ns is the total number of the off-diagonal apparent resistivities and phases to be obtained and ns is the number of receivers. Mesh refinement iteration stops when either the given maximum level of refinement is reached or both the rms and the maximum of the relative difference δ p fall below pre-specified thresholds, which are chosen to be per cent for all model tests in this paper. 4 VALIDATION OF THE ADAPTIVE FINITE ELEMENT CODE To validate the new adaptive finite element algorithm, we examined a -D model and compared the computed results with the analytical solution. The test model consists of km of.3 -m sea water, a -m seafloor and a 5-km thick anisotropic layer at 7 km depth. The resistivity tensor of the anisotropic layer is given by the principal resistivities ρ x /ρ y /ρ z = // -m with a strike angle α s = 3. We calculated MT responses on the seafloor for periods between and 5 s, five periods per decade. A model domain of km width and km height was initially discretized

4 FE modelling of -D MT fields in general anisotropic media (a) 5 (b) ρ a ( analytic FE * * * Phase (degree) 5 5 analytic FE * * * * ρ a error in percent (c) Phase error in degree (d) T(s) T(s) Figure. Apparent resistivities (a) and phases (b) on the seafloor for a -D model with a sea water layer and an electrical anisotropy layer. The solid lines indicate analytical solutions and asterisks indicate FE results obtained on a refined grid, which was scaled by using s period responses. The relative errors in apparent resistivity (c) and the absolute error in phase (d). into 74 triangular elements with 89 vertices. Adaptive refinement iteration was set to refine 5 per cent of elements with largest error indicator, which was computed at a period of s. The solution converged to a rms of.8 per cent and maximum relative difference for off-diagonal MT responses of.5 per cent after 35 mesh refinements. The final mesh, consisting of vertices and 336 triangular elements was also used to calculate the MT responses at the remaining periods. The full components of both the apparent resistivity and the phase obtained by the FE method (symbols) on the s optimized grid are shown in Figs (a) and (b). The -D analytical solution (solid line) is also shown for comparison. It is seen that the FE algorithm provides accurate results over the whole range of periods considered here. The relative deviation between the numerical and analytical apparent resistivity is less than per cent at periods from s to 3 3 s (Fig. c) and the error in phase is less than at periods from s to 4 4 s (Fig. d). At periods larger than few thousand seconds, however, the misfit becomes large and the accuracy of the numerical results decreases. This indicates that the refined grid, scaled by using s responses, is not the best one for calculating MT responses at long periods. Adaptive grid refinement iteration was done once again, but at a period of 4 s. The rms and the maximum relative difference of MT responses converged to.4 and.7 per cent, respectively, after 34 grid refinements. In Figs (a) and (b), errors between the -D analytical solution and the numerical results computed by using the 4 s optimized grid are shown. At periods larger than 3 s, the relative error in the apparent resistivity and the absolute error in phase are less than per cent and.5, respectively. At periods shorter than few hundred seconds, however, errors are large and the accuracy of the numerical results is poor. The above numerical examples show that finite element grids have the period dependence and the single grid for ensuring accurate solutions for the whole range of periods does not exist. To obtain accurate FE solution, the ideal procedure is to use the adaptive refinement for each period. However, this method requires more computer time. Numerical experiments show that the FE algorithm can provide very accurate results when a refined grid is used to compute the MT response for one decade of periods on both sides of the refined period. We used the and 4 s refined grids to compute MT responses at periods from to 3 s and from 3 to 5 s, respectively, and the deviation to the analytical solution are shown in Figs (c) and (d). 5 NUMERICAL EXAMPLES 5. A table mountain The first example shows a simplified bathymetry feature, a table mountain, and has been specifically chosen to make a meaningful comparison possible between the present adaptive FE code and the FD solution, restricted to mesh-aligned boundaries. We consider a three-layer model shown in Fig. 3. The upper layer is km thick and has a resistivity of.3 -m, which approximates sea water. The third layer is an isotropic half-space with a resistivity of -m. Between them is a 6-km-thick anisotropic layer. On the top of the anistropic layer, there is a block elevation of the width of km and height of. km. An isotropic dipping slab with a resistivity of -m is embedded into the anisotropic layer. The resistivity of the anisotropic layer can be described by a resistivity tensor or three principal resistivities and three Euler angles. Although our code allows for models with varying anisotropic slant, we keep the slant constant of in the numerical examples, considering the complexity of the MT responses due to strong coupling between the anisotropic strike and slant. For all the model calculations in this section, the horizontal resistivities of the anisotropic layer (ρ x and ρ y ) remain constant and are set to be -m, whereas the vertical resistivity ρ z and the strike angle and dip angle may vary to demonstrate the influence of anisotropy on the MT response. First, we study the effect of the vertical resistivity of the anisotropic layer on MT responses. For layered conductors excited by a normal plane electromagnetic wave, the vertical resistivity is Journal compilation C 8 RAS

5 946 Y. Li and J. Pek ρ a error in percent (a) computed on the 4 s optimized grid Phase error in degree (b) computed on the 4 s optimized grid ρ a error in percent (c) computed on both s and 4 s optimized grids T(s) Phase error in degree (d) computed on both s and 4 s optimized grids T(s) Figure. Errors between the analytical solution and FE results, which were obtained on a 4 s refined grid in (a) and (b) or on both and 4 s optimized grids in (c) and (d). 7 ρ ρ x = ρ.3 Ωm y = Ωm z = variable Dip angle = variable Strike angle = variable Ωm -. km km Ωm.5 km km km 3km y(km) z(km) Figure 3. A three-layer model: the upper layer and the third layer are sea water and an isotropic half-space, respectively. An anisotropic layer is in between and there is a block seafloor elevation. An isotropic dipping slab is embedded into the anisotropic layer. This model is used to demonstrate anisotropy effects upon MT responses. a principally unresolvable parameter of the medium. It can be only detected through its effect on non-uniform secondary fields arising due to scattering on lateral inhomogeneities within the conductor. It is undoubtedly of great importance to understand how the scattered fields may behave at contacts between the sea water and stacked layers of possibly anisotropic sediments beneath the sea bottom under the influence of both small-scale bathymetric distorters and extensive deeper inhomogeneities. Fig. 4 shows the apparent resistivity (top row) and phase (bottom row) curves along the seafloor for two vertical resistivities (ρ = and -m) at two periods of and s. The dashed lines indicate the MT response computed by the FE scheme and the asterisks indicate the results obtained by using the previously developed FD code (Pek & Toh ), and the fit is excellent. From Fig. 4, one can see that the TE mode MT curves (i.e. ρ and ) are independent of the vertical resistivity and are not affected by the vertical anisotropy, whereas the TM mode curves (i.e. ρ and )areaffected considerably by the vertical resistivity. The topography effect is observed and both modes are clearly affected by bathymetry. TE mode apparent resistivity is smaller above the elevation, whereas

6 FE modelling of -D MT fields in general anisotropic media 947 apparent resistivity( m).5.5 ρ FE ρ z FD * * T = s.5.5 ρ T = s.5.5 ρ T = s m seafloor.5.5 ρ T = s Phase (degree) Φ T = s Φ T = s Φ T = s Φ T = s Figure 4. Apparent resistivity (top row) and phase (bottom row) curves at two periods of and s, along the seafloor for two vertical resistivity (ρ = and -m) in the anisotropic layer from Fig. 3. The dashed line and the asterisk indicate FE and FD results, respectively. TM mode apparent resistivity is bigger above the elevation and has huge jumps at the edges of the elevation. The weak asymmetry of the MT curves with respect to the axis y = is due to the effect of the deep dipping slab. Though the anomalous MT response due to the seafloor table mountain is rather dramatic, it is only little influenced by the vertical anisotropy of the sea bottom sediments. The relative change of the TM apparent resistivity above the centre of the mountain due to the increase of the vertical resistivity from to -m does not exceed per cent for the two periods considered, and the maximum phase difference is less than, immediately beyond the block scarps. With the resistive dipping slab introduced, the differences between the TM apparent resistivities still remain small, not exceeding per cent, Apparent resistivity ( m) 6 FE α s FD * 4 3 ρ T = s 4 4 ρ T = s 4 4 but the impedance phases vary by more than 5 greater/smaller above/beyond the block, respectively, for the sediments with the increased vertical resistivity of -m, compared with the isotropic case. For the model under consideration, the highly conductive sea water environment generally decreases the effect of the vertical anisotropy of the sediments. The MT responses, analogous to those studied above for a model with the sea layer removed, are generally stronger, particularly above the block. Next, we consider a general anisotropic case, in which the nonzero anisotropic strike and dip are incorporated into the above vertical anisotropic model. This model simulates a case of sedimentary layers dipping under an arbitrary angle with respect to the structural strike of the model. Fig. 5 illustrates the apparent resistivity (top 6 ρ T = s 4 seafloor m 4 4 ρ T = s Phase (degree) Φ T = s Φ T = s Φ T = s Φ T = s Figure 5. Apparent resistivity (top row) and phase (bottom row) curves at two periods of and s, along the seafloor for various anisotropy strike angles in the anisotropic layer from Fig. 3. The anisotropy dip is fixed and equals 3. The dashed line and the asterisk indicate FE and FD results, respectively. Journal compilation C 8 RAS

7 948 Y. Li and J. Pek - y (km).3 Ωm. km km km km ρ ρ x = y = Ωm ρz = Ωm Ωm 3km α d = variable.5 km 7 Ωm z (km) Figure 6. A -D model similar to that one in Fig. 3. A triangular block elevation is located on the seafloor. row) and phase (bottom row) curves along the seafloor for various anisotropy strike α s, and the anisotropy dip is fixed and equals 3. The dashed line and the symbol asterisks indicate the MT responses obtained by using the FE and FD codes, respectively. One can see that MT responses for both the TE and TM modes are now affected by the anisotropy, and generally non-vanishing secondary (diagonal) impedances are generated as well, in this case. Note that the diagonal components of the apparent resistivities for both the table mountain model and the sea hill model discussed later are very small (less than.3 -m) and are not plotted. By varying the anisotropy strike α s, the principal directions of the equivalent horizontal resistivities ρ h x and ρh y change in that they follow the anisotropy strike direction. This explains the relatively large differences between the MT curves for different values of α s. For the above finite element modelling, a model domain of km width and km (including km in the air) height was initially discretized into 38 triangular elements with 74 vertices. A total of 5 receivers is situated along the seafloor between y = 5 and 5 km. Adaptive refinement iteration was set to refine 5 per cent of elements with largest error indicator calculated at a period of s. For most models, the rms and the maximum relative difference converged to less than per cent, after about 5 grid refinements. For the general anisotropy model with α s = α d = 3, for instance, the rms and the maximum relative difference converged to. and.6 per cent, respectively, after 5 grid iterations, resulting in triangles and 7 vertices. The grid refinement iteration took s and the refined grid required 88 s to compute the MT solutions at both periods of and s. For the finite difference calculations, we used a grid with 8 and 3 mesh steps in the horizontal and vertical direction, respectively. The air layer was formed of 4 steps, so, 9 vertical steps were there in the conductor. The smallest steps were 5 and m in the horizontal and vertical direction, respectively. The extent of the whole model was about 6 km horizontally and about 8 km vertically (from those, about 5 km in the air). With these grid parameters, it took about s to compute two periods MT responses of the general anisotropy model (PC Intel Xeon 3 GHz). 5. A sea hill In this subsection, we studied the performance of both the FE and FD codes for models with oblique interfaces. Fig. 6 shows a -D model, which is similar to the previous -D model in Fig. 3, but with the seafloor block elevation replaced by a triangle elevation. For the FE modelling, the coarse initial grid consists of 63 triangular elements with 654 vertices. Adaptive refinement iteration was set to refine 5 per cent of elements with largest error indicator, which was calculated at T = s. For the model with α d = 3, the rms and the maximum relative difference converged to. and.5 per cent, respectively, after 9 grid iterations, resulting in a grid of 5 7 elements and 6 6 vertices. The FE simulation including 9 grid refinements took 38 s compute time. The FD model was designed in the same mesh as the table mountain model above, with the size of mesh being 8 3 steps in the horizontal and vertical directions, respectively, covering a model area of 6 8 km. The hill is approximated in a mesh subdomain of 4 horizontal vertical steps, with the size of each mesh cell being.5. km. To check the accuracy of the modelling results, the mesh was further refined by a factor of two in both directions, with the hill geometry unchanged. Refining the mesh increases details in the fields along the treads of the steps, which change considerably due to the nearness of high contrasts at the risers of the steps. A cascade of still finer oscillations can be obtained if the mesh is further refined, producing a channel with an average curve, well consistent with the FE solution. Fig. 7 shows the apparent resistivity and the phase for various anisotropy dip angles. The both TE and TM mode MT responses are significantly distorted by the seafloor elevation. The TE mode apparent resistivity and phase computed by the FE scheme agree very well with those by the FD code. The TM mode MT responses fit well only where the seafloor is flat. On the slopes of the bathymetry elevation, the apparent resistivity ρ and phase from the FD code are zig-zagged and oscillating around those computed using the FE code, which is a consequence of the FD algorithm allowing us to approximate the seafloor slope by the staircase steps only. Under such a structural approximation, each step represents a contact of

8 FE modelling of -D MT fields in general anisotropic media 949 ρ ( m).5.5 sea hill, Varying anisotropy dip (α d ) FE α d FD * 3 * 6 * 9 * seafloor ρ ( m) Period T= s m φ (degree) φ (degree) Figure 7. Apparent resistivities (top row) and phases (bottom row) at a period of s along the seafloor for various anisotropy dip angles in the anisotropic layer from the sea hill model in Fig. 6. The dashed lines and the asterisks indicate FE and FD results, respectively. The TM mode FD results are oscillating around the slope of the elevation. highly electrically contrasting media and produces a discontinuity in the MT impedances. 5.3 A volcano model with strong onshore topography We finally present a volcano example that demonstrates the capabilities of the adaptive unstructured mesh FE algorithm for modelling electrical anisotropy structures with strong topography undulations. We consider hypothetical electrical structures roughly corresponding to those of the Mount Somma Vesuvius volcano complex, which has been studied by direct current, self-potential and MT methods (Maio et al. 998; Manzella et al. ). We chose a NE SW profile crossing the volcano and designed a -D model shown in Fig. 8. A shallow sea layer is to the SW. The topography undulates roughly in the central part of the volcano, and the maximum gradient of the slope reaches 45. The shallower part of the volcano is modelled as horizontally anisotropic to emulate possible effects of factors related to some shallower volcanic structures, especially radial fractures radiating from the centre of the volcano and possible anisotropy of the hydraulic permeability controlling hydrothermal flows beneath the volcano (e.g. Bonafede & Cenni 998). The resistivity tensor of the shallower anisotropy layer is given by the principal resistivities and a strike angle α s andtheyaresettobeρ x /ρ y /ρ z = 4//4 -m and α s = 75 on the left-hand flank and ρ x /ρ y /ρ z = 5/6/5 -m and α s = 75 on the right-hand flank of the volcano. The integrated electrical and electromagnetic survey (Maio et al. 998) and the MT investigation (Manzella et al. ) showed the existence of a shallow largely conductive zone, closely in correspondence with the Crater of the Somma-Vesuvivs. This conductive zone might be also horizontally anisotropic due to some preferential flow direction of the hot water, and its resistivities are set to ρ x /ρ y /ρ z =.5//.5 -m and anisotropy to α s = 75 on the left-hand flank and + 75 on the right-hand flank of the volcano. By using 3-D forward modelling, Manzella et al. () studied the influence on MT responses of a deep magma chamber body with a resistivity of -m and base area of km. We tested similar -D bodies and found that their effects on MT responses are hardly seen. According to our -D simulations, for the magma chamber body to be sensed by the surface data, it must be shallower by at least km and must have a larger cross-section than that studied by Manzella et al. (). We calculated the MT responses at periods from 3 to 3 s by using the FE code. A total of receiver sites was located at a range from.5 to.7 km along the topography. A model domain of km width and km height (about km in the air) was initially discretized into 464 triangular elements with 346 vertices. The central portion of the starting mesh is shown in Fig. 9(a). Fine elements were generated near coast to conform the thin sea layer. Adaptive refinement iteration was set to refine 5 per cent of elements with the largest error indicator, which is computed at periods of., and s. The resulting refined grids were used to compute MT responses in period intervals of.., Journal compilation C 8 RAS

9 95 Y. Li and J. Pek Figure 8. A -D volcano model. The near-suface structures of the volcano are horizontally anisotropic with resistivities ρ x /ρ y /ρ z = 4//4 -m and anisotropy strike a s = 75 on the left-hand flank and ρ x /ρ y /ρ z = 5/6/5 -m and a s = 75 on the right-hand flank. The conductive zone is also horizontally anisotropic with resistivities ρ x /ρ y /ρ z =.5//.5 -m and anisotropy strike a s = 75 on the left-hand flank and and a s = 75 on the right-hand flank. A conductive body ( -m) is embedded into a resistive basement ( -m).. and s, respectively. When using a period of. s, the rms value and maximum relative difference converged to. and.8 per cent, respectively, after 6 grid refinement and the iteration process was terminated. The final mesh contains 7 7 elements with 35 4 vertices, and its central portion is shown in Fig. 9(b). The mesh refinement is concentrated in the regions around the topography at y = to 3 km, where MT sites are located and the topography changes roughly. Additional mesh refinement was centred in regions where geometry of electrical bodies changes roughly. Although not shown here, the resulting refined grids from and s periods have similar features. Fig. shows full components of MT impedance on the lefthand flank of the volcano (y = 4 km). The solid lines denote the responses for the anisotropic near-suface structures without the magma chamber body, asterisks indicate the corresponding responses of the equivalent isotropic structures and circles are for both the anisotropic near-surface and the magma chamber body. The influence of the near-surface anisotropy on the MT responses are clearly seen at short periods. The diagonal component ρ is significant and huge at all periods, whereas ρ is significant only at short periods from. to about.3 s at y = 4 km. The effect of the magma chamber body is clearly seen for the - and -components. Fig. shows the apparent resistivity and phase curves along the volcano topography at periods of.5 and s. The topography effect on the MT responses is clearly observed. The size of variations correlated with the topography features seems to be comparable with most of the differences due to the shallow anisotropy introduced in our hypothetical model. Thus, inaccuracies in the topography model may effectively mask small anisotropy that would qualitatively conform to assumed anisotropies in the hydraulic permeabilities (e.g. Bonafede & Cenni 998). 6 CONCLUSIONS We have presented an adaptive finite element solver for MT forward problem in -D generally anisotropic media, aiming, in particular, at improved rendering of arbitrarily shaped structural boundaries, rough topographies and bathymetries. Comparing results of the adaptive FE modelling with those of the commonly used -D FD code by Pek & Verner (997) and its extension by Pek & Toh (), we can conclude that both the FD and FE schemes can give results of practically the same accuracy for model structures without oblique topography/bathymetry undulation. However, the FD code for -D anisotropic MT forward modelling is not suitable for models with oblique boundaries due to the forced mesh alignment of the boundaries and a staircase approximation of the topography/bathymetry features. The mesh design is fully manual in the FD code; so, the numerical accuracy of the model results is not independent of the user s experience in constructing the mesh in complex, multi-period and often multi-scale electrical models. The use of the unstructured mesh in the FE approach enables us to model arbitrarily complex -D structures. Adaptive finite element methods provide a powerful approach for numerical modelling of complex problems and can automatically improve mesh design and offer reliable solution at a reasonable computational cost. Although our numerical examples mainly aimed at showing the effect of the adaptive FE code in practical situations, some general

10 FE modelling of -D MT fields in general anisotropic media 95 Figure 9. The central portion of the refined meshes using. s data for the model in Fig. 8. (a) Starting mesh. (b) Final mesh generated using. s data after 6 refinement iteration. features regarding the influence of anisotropy on the MT data could be inferred as well. Our modelling results show a clearly nonnegligible effect of the vertical electrical anisotropy in seafloor MT data. This factor is often disregarded in MT studies, mainly because of complete insensitivity of MT data to the vertical anisotropy of purely layered media. In the seafloor conditions, the primary, inducing magnetic field may be largely attenuated, and the MT responses rely to a considerable degree on the secondary, scattered fields. In a purely vertically anisotropic structure, only TM mode responses are affected by the vertical anisotropy and only in some close vicinity of laterally non-uniform domains of the model. We observed that this effect is non-negligible in our model setting, with the anisotropy concentrated in the sedimentary layer beneath the sea bottom (Fig. 4), and the anisotropy effect can be sensed laterally at a distance of a few widths of the anomalous inclusions in the ρ and curves. Vertical anisotropy of isolated inhomogeneities seems to be of almost negligible effect, at least for bodies used in simulating hypothetical magma chamber bodies beneath some volcanoes (Fig. ). In a general anisotropy case, MT responses of both the TE and TM-modes are affected by the anisotropy, and generally nonvanishing diagonal impedances are generated as well. In our seafloor models, the secondary impedances were very small as a rule, of magnitudes not particularly useful for practical interpretations. In the volcano model with rough topography, however, considerable secondary impedances appear due to the anisotropy (Figs and ), which are further modulated by topographic undulations and lateral non-uniformities in the underground. These effects are highly model-specific, but, by comparing the responses for isotropic and anisotropic models at different periods (Fig. ), they may be Journal compilation C 8 RAS

11 95 Y. Li and J. Pek Figure. Full components of MT impedance on the left-hand flank of the volcano (y = 4 km) in Fig. 8. Solid lines denote the responses for anisotropic near-surface structures without the magma chamber body, asterisks indicate the corresponding responses of the equivalent isotropic structures and circles are those for both the anisotropic near-surface and the magma chamber body. decomposed into an anisotropy induction part and a generally directed (i.e. not aligned with the structural strike) static distortion due to the interaction between the anisotropy and topography. ACKNOWLEDGMENTS YL acknowledges funding support from BP America. The contribution of JP would not have been possible without a financial assistance of the Grant Agency of the Acad. Sci. Czech Rep., contract No. IAA34, and the Czech Sci. Found., contract No. 5/6/557. Several subroutines were originally written by Kerry Key when he and YL collaborated (Li & Key 7). We thank Kerry Key for the Matlab graphical model design interface. We also thank Chester Weiss for the matrix-free QMR code (Weiss ) and Jonathan Shewchuk for the grid generation code Triangle (Shewchuk ). Thanks are also due to Oliver Ritter, Ralph-Uwe Börner and another anonymous referee for their constructive comments. REFERENCES Ainsworth, M. & Oden, J.T.,. A Posteriori Error Estimation in Finite Element Analysis, John Wiley, New York. Baba, K., Chave, A.D., Evans, R.L., Hirth, G. & Mackie, R.L., 6. Mantel dynamics beneath the east pacific rise at 7 s: insights from the mantel electromagnetic and tomography (MELT) experiment, J. geophys. Res.,, B, doi:.9/4jb3598 Bonafede, M. & Cenni, N., 998. A porous flow model of magma migration within Mt. Etna: The influence of extended sources and permeability anisotropy, J. Volc. Geotherm. Res., 8, Brasse, H., Li, Y., Kapinos, G., Eydam, D. & Muetschard, L., 5. Uniform deflection of induction vectors at the South Chilean continental margin: a hint at electrical anisotropy in the crust, in Protokoll Kolloquium Electromagnetische Tiefenforschung, pp. 8 87, eds Ritter O. & Brasse, H., Haus Wohldenberg, Halle, Germany, DGG. Evans, R.L. et al., 999. Asymmetric electrical strucure in the mantle beneath the East Pacific Rise at 7 S, Science, 86,

12 FE modelling of -D MT fields in general anisotropic media 953 Figure. Apparent resistivities and phases along the topography at periods of.5 s and s for the model in Fig. 8. Evans, R.L., Hirth, G., Baba, K., Forsyth, D., Chave, A. & Mackie, R., 5, Geophysical evidence from the MELT area for compositional controls on oceanic plates, Nature, 437(756), Franke, A., Börner, R.-U. & Spitzer, K., 7. Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography, Geophys. J. Int, 7, Han, W., 5. A Posteriori Error Analysis via Duality Theory: With Applications in Modeling and Numerical Approximations, Springer Science, New York. Hou, J., Mallan, R. & Torres-Verdin, C., 6. Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials, Geophysics, 7, Key, K. & Weiss, C., 6. Adaptive finite-element modeling using unstructured grids: the D magnetotelluric example, Geophysics, 7(6), G9 G99. Li, Y.,. A finite-element algorithm for electromagnetic induction in two-dimensional anisotropic conductivity structures, Geophys. J. Int., 48, Li, Y. & Key, K., 7. D marine controlled-source electromagnetic modeling, part : an adaptive finite-element algorithm, Geophysics, 7(), WA5 WA6. Maio, D., Mauriello, R.P., Patella, D., Petrillo, Z., Piscitelli, S. & Siniscalchi, A., 998. Electric and electromagnetic outline of the Mount Somma- Vesuvius structural setting, J. Volc. Geotherm. Res., 8, Manzella, A., Volpi, G. & Zaja, A.,. New magnetotelluric soundings in the Mt. Somma-Vesuvius volcano complex: preliminary results, Annali Di Geofisica, 43, Ovall, J.S., 4. Duality-based adaptive refinement for elliptic PDEs, PhD thesis, University of California, San Diego. Ovall, J.S., 6. Asymptotically exact functional error estimators based on superconvergent gradient recovery, Numerische Mathematik, (3), Pek, J. & Toh, H.,. Numerical modeling of MT fields in D anisotropic structures with topography and bathymetrey considered, in Protokoll Kolloquium Elektromagnetische Tiefenforschung, pp. 9 99, eds Hoerdt A. & Stoll, J., Altenberg, Germany, DGG. Journal compilation C 8 RAS

13 954 Y. Li and J. Pek Pek, J. & Verner, T., 997. Finite-difference modeling of magnetotelluric fields in two-dimensional anisotropic media, Geophys. J. Int, 8, Shewchuk, J.R., 997. Delaunay refinement algorithms mesh generation, PhD thesis, School of Comupter Science, Carnegie Mellon University, Pittsburgh, Pennsylvania (Technical Report CMU-CS-97-37). Shewchuk, J.R.,. Delaunay refinement algorithms for triangular mesh generation, Comput. Geometry (Theory and Applications),, 74. Wang, T. & Fang, S.,. 3D electromagnetic anisotropy modeling using finite differences, Geophysics, 66, Wannamaker, P.E., Jiracek, G.R., Stodt, J.A., Caldwell, T.G., Gonzalez, V.M., McKnight, J.D. & Porter, A.D.,. Fluid generation and pathway beneath an active compressional orogen, the New Zealand southern alps, inferred from magnetotelluric data, J. geophys. Res., 7(B6),7, doi:.9/jb86. Weidelt, P., 999. Three-dimensional conductivity models: implications for electrical anisotropy, in Three-diemnsional Electromagnetics, pp. 9 37, eds Oristaglio, M. & Spies, B., Soc. Expl. Geophys. Weiss, C.J.,. A matrix-free approach to solving the fully 3D electromagnetic induction problem, 7th Annual Internat. Mtg., Soc. Expl. Geophys. (Expanded Abstracts), (), Weiss, C.J. & Newman, G.A.,. Electromagnetic induction in a generalized 3D anisotropic earth, Geophysics, 67, 4 4. APPENDIX A: DUAL ERROR ESTIMATE WEIGHTING The dual error estimate weighting uses the solution to a dual problem to bias refinement towards elements that affect the solution at the EM receiver locations and enables the computation of asymptotically exact solutions to the partial differential equations. Consider a functional G, which is some measure of the solution error u u h, where u is the true solution of the partial differential equation and u h is the finite element approximation. The dual problem of eq. (8) reads B (w, v) =, (A) for w in which B is a dual or adjoint operator and is defined as B (w, v) = B(v, w). We then have G(u u h ) = B (w, u u h ) = B(u u h,w) = B(u u h,w w h ), (A) where w and w h are the true solution and the FE solution of the dual problem, respectively. In the derivation of the last term on the righthand side of eq. (A), we have used the orthogonality property B(u u h, w h ) = F(w h ) B(u h, w h ) =. Provided with FE solutions u h and w h, the right-hand side of eq. (A) can be used to compute the equivalent solution to the error functional G: B(u u h,w w h ) = α (u u h ) (w w h )d, (A3) where the contribution of the last two terms in eq. (8) has been negelected. Approximating the gradient terms by (u u h ) (R I ) u h, (w w h ) (R I ) w h (A4) yields B(u u h,w w h ) α(r I ) u h (R I ) w h d. (A5) Eq. (A5) is an approximation of the error functional G, which can be computed using u h and w h, the solutions of the primal and dual problem, respectively. From (A5), we define the error indicator as ˆη e = η e η e. (A6)