Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X38420130219Enhancing the resolving power of 2D least-squares inversion of magnetotelluric dataEnhancing the resolving power of 2D least-squares inversion of magnetotelluric data1171303021010.22059/jesphys.2013.30210FAMajidJamieBehroozOskooi0000-0003-3065-194XJournal Article19700101This paper presents results of applying a new approach on 2D inversion of Magnetotelluric (MT) data in order to enhance resolution and stability of the inversion results. Due to non-linearity and limited coverage of data acquisition in an MT field campaign, minimizing the error by linearization of the problem in least squares inversion usually leads to an ill-posed problem. In general, an inverse problem is unstable, ill-posed and is characterized by non-uniqueness (Tikhonov et al., 1998). The concept of ill-posed problems goes back to Hadamard (1923). He defined a problem to be ill-posed if the solution is not unique or if it is not a continuous function of the data i.e., arbitrarily small perturbations in the input data can cause great changes in the solution. Hence, in order to stabilize the problem and come to a stable solution, further information should be incorporated. In order to acquire reasonable geoelectrical models, regularization of the problem by imposing definite constraints is necessary. Determination of a suitable Lagrangian multiplier in order to balance minimization of error and model roughness could be a useful approach to achieve both the resolution and the stability in inversion.
In order to achieve both the resolution and the stability in least-squares inverse modelling in our study, an intermediate value of the Lagrangian multiplier must be chosen. Too large or small Lagrangian multipliers yield to undesirable effects on resolution and stability. In this paper, the regularization parameter is set by a value from log-linear interpolation (Yi et al., 2003).
Where, , and , are minimum and maximum for Lagrangian multipliers and spread function respectively. The regularization parameter can be set optimally by the spread function of the ith model parameter which is defined by the parameter resolution matrix R. The spread function shows how much the ith model parameter is not resolvable and is written as bellow:
Where, M is the total number of inversion parameters, is a weighting factor defined by the spatial distance between the ith and jth model parameters, and is a factor which indicates whether the constraint or regularization is imposed on the ith parameter and its neighboring parameters. An alternative to varying the Lagrangian multiplier as the iterations proceed is to use the spatially varying Lagrangian multiplier (Sasaki, 1989). Hence, varying the Lagrangian multiplier by trial and error is preferred to get resolution and stability. Small regularization parameters mean higher resolvable inversion blocks in parameter resolution analysis sense (Menke, 1989).
We tested the capability of the Active Constraint Balancing (ACB) approach (Yi et al., 2003) in enhancing the resolving power of least-squares inversion results by applying it on 2D synthetic MT data generated from forward modeling code of Geotools-MT for simple models of the earth and then on the field data. Using ACB approach, the rms error and data misfit is much less than the conventional approach with fixed Lagrangian multiplier, which leads to higher resolving power and the stability of the inversion results. The inversion code which was used in this paper (Lee et al., 2009) consists of finite element for computing 2D MT model responses, and smoothness-constrained least-squares inversion. By comparing the resistivity sections, the anomalous object can be seen much clearer and distinct in the case of ACB approach. This enhancement in the resolution could be well interpreted as the result of using varying Lagrangian multipliers in the smoothness-constrained least-squares inversion using ACB approach.This paper presents results of applying a new approach on 2D inversion of Magnetotelluric (MT) data in order to enhance resolution and stability of the inversion results. Due to non-linearity and limited coverage of data acquisition in an MT field campaign, minimizing the error by linearization of the problem in least squares inversion usually leads to an ill-posed problem. In general, an inverse problem is unstable, ill-posed and is characterized by non-uniqueness (Tikhonov et al., 1998). The concept of ill-posed problems goes back to Hadamard (1923). He defined a problem to be ill-posed if the solution is not unique or if it is not a continuous function of the data i.e., arbitrarily small perturbations in the input data can cause great changes in the solution. Hence, in order to stabilize the problem and come to a stable solution, further information should be incorporated. In order to acquire reasonable geoelectrical models, regularization of the problem by imposing definite constraints is necessary. Determination of a suitable Lagrangian multiplier in order to balance minimization of error and model roughness could be a useful approach to achieve both the resolution and the stability in inversion.
In order to achieve both the resolution and the stability in least-squares inverse modelling in our study, an intermediate value of the Lagrangian multiplier must be chosen. Too large or small Lagrangian multipliers yield to undesirable effects on resolution and stability. In this paper, the regularization parameter is set by a value from log-linear interpolation (Yi et al., 2003).
Where, , and , are minimum and maximum for Lagrangian multipliers and spread function respectively. The regularization parameter can be set optimally by the spread function of the ith model parameter which is defined by the parameter resolution matrix R. The spread function shows how much the ith model parameter is not resolvable and is written as bellow:
Where, M is the total number of inversion parameters, is a weighting factor defined by the spatial distance between the ith and jth model parameters, and is a factor which indicates whether the constraint or regularization is imposed on the ith parameter and its neighboring parameters. An alternative to varying the Lagrangian multiplier as the iterations proceed is to use the spatially varying Lagrangian multiplier (Sasaki, 1989). Hence, varying the Lagrangian multiplier by trial and error is preferred to get resolution and stability. Small regularization parameters mean higher resolvable inversion blocks in parameter resolution analysis sense (Menke, 1989).
We tested the capability of the Active Constraint Balancing (ACB) approach (Yi et al., 2003) in enhancing the resolving power of least-squares inversion results by applying it on 2D synthetic MT data generated from forward modeling code of Geotools-MT for simple models of the earth and then on the field data. Using ACB approach, the rms error and data misfit is much less than the conventional approach with fixed Lagrangian multiplier, which leads to higher resolving power and the stability of the inversion results. The inversion code which was used in this paper (Lee et al., 2009) consists of finite element for computing 2D MT model responses, and smoothness-constrained least-squares inversion. By comparing the resistivity sections, the anomalous object can be seen much clearer and distinct in the case of ACB approach. This enhancement in the resolution could be well interpreted as the result of using varying Lagrangian multipliers in the smoothness-constrained least-squares inversion using ACB approach.https://jesphys.ut.ac.ir/article_30210_926fd3964a0fd09939559150b418b968.pdf