A non-linear truncated SVD variance and resolution analysis of two-dimensional magnetotelluric models

Abstract

S U M M A R YA novel approach to assess variance and resolution properties of 2-D models of electrical resistivity derived from magnetotelluric measurements is presented. Based on a truncated singular value decomposition (TSVD) scheme on a local subspace, it partly takes the nonlinearity of the inverse problem into account. The TSVD resolution and variance analysis is performed on a single cell at a time. A variance threshold is selected and the resulting model resolution is determined. As an improvement over existing schemes, non-linear semi-axes are introduced to describe the non-linear confidence surface in the directions of the model eigenvectors and they replace the inverse singular values entering into the standard expression of model variances. The model variance of the cell considered is estimated from the sum of squares of the non-linear semi-axes up to the given variance threshold. This, in turn, gives the truncation level of the TSVD and the row of the model resolution matrix belonging to the considered cell can be computed from the model eigenvectors of the TSVD. The information contained in the resolution matrix is condensed to easily comprehensible measures like the centre of resolution and horizontal and vertical resolution lengths.The validity of our non-linear model variance and resolution estimates is tested with a most-squares technique which gives an improved estimate of model variability.A synthetic model with a conductive block in a homogenous half-space is analysed. TSVD analyses for model cells on the upper edge of the block and outside the block illustrate how the truncation process works. Typically, the linear and non-linear semi-axes are almost equal up to a certain singular value number, after which the non-linear semi-axes increase much less than the linear semi-axes. This important result indicates that the resolution of 2-D magnetotelluric models is significantly better than previously suggested by linear schemes for the computation of model variance and resolution properties. A field example from the Skediga area (Sweden) shows that the electrical resistivity distribution of sand and gravel formations which are only laterally bounded by conductive clay lenses is relatively well resolved whereas there is little resolution for the transition between the sand and gravel layer and the basement under a clay cover.

Two-Dimensional Magnetotelluric Modelling of Ore Deposits: Improvements in Model Constraints by Inclusion of Borehole Measurements

“…This latter procedure would have to be repeated iteratively with further adjustment of the considered model parameter until the misfit functional assumes the value that corresponds to minus or plus one standard deviation of the model parameter (cf. Kalscheuer and Pedersen 2007;Kalscheuer et al 2010).…”

“…Since the smoothness-constrained most-squares inversion seems to work unreliably for data sets containing both surface and borehole measurements, the model error estimates from linearisation could not be compared to nonlinear estimates. Future attempts to compute nonlinear model error estimates may use a most-squares inversion based on singular value decomposition (Kalscheuer and Pedersen 2007), or an iterative procedure where the resistivity of the cell under investigation is fixed at a value smaller or larger than that of the preferred inversion model, and the remaining model parameters are allowed to vary freely in a subsequent inversion. This latter procedure would have to be repeated iteratively with further adjustment of the considered model parameter until the misfit functional assumes the value that corresponds to minus or plus one standard deviation of the model parameter (cf.…”

“…2), it can be expected that the responses in the vertical magnetic and electric transfer function Z zx and T y , respectively, are smaller in amplitude. Further, since the background resistivity predominantly decreases with depth, it can be expected that surface data alone can constrain the model rather well (e.g., Bedrosian 2007;Kalscheuer and Pedersen 2007). In our experimental set-up, 38 surface receivers are evenly distributed along the profile at a station spacing of 200 m with the first receiver at 800 m along the line.…”

“…Since Kalscheuer et al's (2010) algorithm employs logarithmic resistivities of the 2D cells, the errors on actual cell resistivities are factors f corresponding to resistivity ranges q=f ; f q ½ . Meju and Hutton (1992), Meju (1994), Kalscheuer and Pedersen (2007) and Kalscheuer et al (2010) demonstrated that verification of linearised model error estimates using nonlinear most-squares inversion is generally desirable.…”

“…For a given model parameter, the resolving kernel r M is computed by scaling the entries of the corresponding row of the model resolution matrix R M by the respective cell areas and, subsequently, we divide the entire resolving kernel by its maximum modulus, thus projecting the resolving kernel into the range of [-1, 1]. We compute the centres of resolution and horizontal and vertical resolution lengths according to Kalscheuer and Pedersen (2007) and indicate these using red crosses in the plots of our resolving kernels. Note that these centres of resolution and horizontal and vertical resolution lengths are appropriate representations of the resolving kernel only when it has a single main lobe.…”