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For an accurate estimation of deep mantle conductivity distributions in 3-D, these topographic effects have to be properly and accurately taken into account in the inversion. In reality, topography variations occur over a wide range of horizontal scales from local (100 m) to regional (1000 km) with amplitudes of only a few kilometers. We can generally expect more accurate solutions by using the incorporation approach described in Li et al. (2008), because inversion solutions obtained using the correction approach depend on the accuracy of the approximation of the sub-seafloor structure. It is, however, neither efficient nor practical to incorporate topographic variations into one forward calculation using finer grid cells, especially when focusing on heterogeneous conductivity structures in the oceanic upper mantle and deeper parts with that are typically on a scale of 100 km or more both in the horizontal and vertical directions.
[FULL] Chave Do Topograph 98
The aim of this paper is to propose an approximate treatment of topography that can be incorporated into 3-D seafloor MT inversion codes to study regional-scale mantle structure. In this study, new techniques are incorporated into the 3-D inversion code, WSINV3DMT (Siripunvaraporn et al., 2005), which is, at present, one of the practical inversions applied to land MT data.
We introduce a general treatment for incorporating topography and bathymetry into the model in Section 2. In Section 3, methods for calculating MT responses at arbitrary points on the undulating seafloor are introduced. The accuracy of the methods described in Sections 2 and 3 is tested in Section 4. In Section 5, a method for calculating sensitivity during inversion is derived. Finally, we apply the method to three kinds of synthetic datasets to verify its performance and then discuss the results in Section 6.
The present problem is how to accurately express seafloor topography (the conductivity boundary) without significantly increasing the computational burden. To avoid using fine grids in the vertical direction, we propose applying a volumetric average of the conductivity for a block that includes the boundary (seafloor), as shown in Fig. 1. When fine topography is available in each block with a horizontal areal resolution of ΔS, the average conductivity of the block is obtained to conserve the horizontal conductance
where σ s and σ c are the electrical conductivities of seawater and crustal rock, and V s and V c are the volumes occupied by seawater and crust in the block, respectively. This treatment allows us to easily design a mesh, regardless of the seafloor topography. It also does not increase the number of blocks in the vertical direction, thus restraining the number of model parameters, which is essential for compact forward calculation and for practical 3-D inversion.
Principle behind the incorporation of seafloor topography. (a) Example topography in rectangular blocks for numerical modeling. Seawater (white region) and crustal rock (dark region) have constant electrical conductivities σ s and σ c , and volumes and V c , respectively in the bold block. (b) The conductivities of the blocks that include both seawater and crustal rock portions, like the bold block, are calculated by conserving horizontal conductance within each block. For example, the conductivity of the bold block is calculated using Eq. (2) and is estimated to be σ h .
The horizontal length scale of the topography included in a model is dependent on its horizontal mesh dimensions, which are usually set to be fine enough to resolve the observation arrays and target structure. Topography with a length scale larger than the horizontal mesh, and having various amplitudes, is efficiently incorporated by the method proposed above. If smaller scale topographic changes exist and their effect is not negligible, we have to divide the horizontal mesh into a finer grid although it increases computational costs. The alternative is an indirect approach, in which such small-scale topographic effects are treated separately and removed from observed EM responses as has been done, for example, by Baba and Chave (2005) for 2-D target structures. Hereinafter, we assume that the responses input to our 3-D inversions are free from the effects of small-scale topography, either because the effect is separately corrected or because it is negligible.
Here, a1,2, b1,2, c1,2, and d1,2 are vectors used to transform the electric fields on the staggered grid to the x - and y-components of the horizontal EM fields at the site for the first (p = 1) and second source (p = 2) polarizations. In this study, the site location on the seafloor can be at any arbitrary position within a block, not just on the block boundaries, because topography is expressed by converting the conductivity as described above. Consequently, the conversion should be considered in the calculation of the transform vectors.
This approximate treatment of topography (hereinafter called ATT) includes expressing the ocean bottom conductivity by volumetric averaging and using interpolation and extrapolation methods to calculate the EM fields at arbitrary observation sites.
where the unit of length is meter. The topography model is discretized every 60 km in the horizontal directions. We calculated MT responses for 11 periods (from 177 to 56,234 seconds) at three points: Site A is located on the flank of a seamount, site B on the flank of a dip, and site C on the flank between two seamounts and two dips. The calculated apparent resistivities and phases are plotted in Fig. 7. The amplitudes of the diagonal components of the impedance tensor are only produced by topographic variations in this test because the conductivity below the seafloor is assumed to be constant and therefore the values of the diagonal elements should be zero when the seafloor is flat. The responses from the two calculations agree within 0.04 on a logarithmic scale of apparent resistivity and 2.9 degrees in the phase of the off-diagonal components, particularly for periods longer than 500 seconds. MT responses at shorter periods show less agreement than those at longer periods since shorter periods are more influenced by the differences between the topographic treatments because the volumetric averaging approximation used in the ATT is less accurate for shorter periods. Although more accurate results can be expected using finer meshes in the vertical direction, in exchange for increased computational cost, this limitation for short periods is not serious, because real marine data deeper than about 5000 m below sea level usually provide good MT responses at periods longer than 500 seconds (Baba et al., 2010). The relatively low accuracy of diagonal elements will not cause a serious problem because their amplitudes are much smaller than those of the off-diagonal elements. In addition, the observation errors in the diagonal elements are typically of the same order as in the off-diagonal elements, and much larger than the differences in Fig. 7. If a 3% relative numerical error is acceptable, one can use MT responses for periods longer than 500 s. If a 10% relative numerical error can be accepted, MT responses at all periods longer than 100 s may be used for inversion. Otherwise, a finer mesh in the vertical direction can be used to get more accurate results, but with increased computational costs. Therefore, we may conclude that incorporating the ATT in the forward calculation of the WSINV3DMT is accurate enough for 3-D inversion.
Three-dimensional view (left) and plan view (right) of the synthetic seafloor topography model. The crosses with labels are the positions where synthetic responses were calculated. Sites A, B, and C are located on the flank of a seamount, on the flank of a dip, and on the flank between two seamounts and two dips, respectively.
Synthetic apparent resistivity and phase at the three locations shown in Fig. 6. Circles, diamonds, and triangles show the responses calculated by the forward part of the WSINV3DMT with approximate treatment of topography (ATT) for sites A, B, and C, respectively. Solid, broken, and dotted lines are the responses calculated by the FS3D algorithm (Baba and Seama, 2002) for sites A, B, and C, respectively.
Here the total performance of the WSINV3DMT with ATT was tested using three synthetic datasets. The three synthetic datasets were generated using the forward part of the WSINV3DMT with ATT code. The first dataset was calculated for a conductive block buried in a half-space below the ocean with constant water depth. The second was calculated for a conductive block buried in a half-space below the ocean with realistic topography, and the third was a checkerboard model below the ocean with realistic topography. All synthetic models were discretized every 60 km in the horizontal direction in the central part of the model domain. The vertical meshes were discretized every 700 m near the seafloor, and the length of the mesh increases exponentially with increasing depth. All the models included seven air layers in the default configuration, and the conductivity values of the seven air layers were fixed in all the inversion calculations. The number of observation sites is 25 where synthetic MT responses are computed. 2ff7e9595c
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