Estimation of regularization parameter by active constraint balancing for 2D inversion of gravity data

Inversion method is very common in the interpretation of practical gravity data. The goal of 3D inversion is to estimate density distribution of an unknown subsurface model from a set of known gravity observations measured on the surface. The regularization parameter is one of the effective parameters for obtaining optimal model in inversion of the gravity data for similar inversion of other geophysical data. For estimation of the optimum regularization parameter the statistical criterion of Akaike’s Bayesian Information Criterion (ABIC) usually used. This parameter is experimentally estimated in most inversion methods. The choice of the regularization parameter, which balances the minimization of the data misfit and model roughness, may be a critical procedure to achieve both resolution and stability. In this paper the Active Constraint Balancing (ACB) as a new method is used for estimating the regularization parameter in two- dimensional (2-D) inversion of gravity data. This technique is supported by smoothness-constrained least-squares inversion. We call this procedure “active constraint balancing” (ACB). Introducing the Lagrangian multiplier as a spatially-dependent variable in the regularization term, we can balance the regularizations used in the inversion. Spatially varying Lagrangian multipliers (regularization parameters) are obtained by a parameter resolution matrix and Backus-Gilbert spread function analysis. For estimation of regularization parameter by ACB method use must computed the resolution matrix R. The parameter resolution matrix R can be obtained in the inversion process with pseudo-inverse  multiplied by the kernel G.
                                                                                                                                            (1)
The spread function, which accounts for the inherent degree of how much the ith model parameter is not resolvable, defined as:
                                                                                                            (2)
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 accounts for whether the constraint or regularization is imposed on the ith parameter and its neighboring parameters. In other words, the spread function defined here is the sum of the squared spatially weighted spread of the ith model parameter with respect to all of the model parameters excluding ones upon which a smoothness constraint is imposed. In this approach, the regularization parameter λ(x,z) is set by a value from log-linear interpolation.
                                                    (3)
where and are the minimum and maximum values of spread function , respectively, and the  and  are minimum and maximum values of the regularization parameter λ(x,z), which must be provided by the user. With this method, we can automatically set a smaller value λ(x,z) of the regularization parameter to the highly resolvable model parameter, which corresponds to a smaller value of the spread function  in the inversion process and vice versa. Users can choose these minimum and maximum regularization parameters by setting variables LambdaMin and LambdaMax. For getting the target an algorithm is developed that estimates this parameter. The validity of the proposed algorithm has been evaluated by gravity data acquired from a synthetic model. Then the algorithm used for inversion of real gravity data from Matanzas Cr deposit. The result obtained from 2D inversion of gravity data from this mine shows that this algorithm can provide good estimates of density anomalous structures within the subsurface.