A Trade-off Solution of Regularized Geophysical Inversion Using Model Resolution and Covariance Matrices

Regularization is necessary for inversion of ill-posed geophysical problems. Appraisal of inverse models is essential to a meaningful interpretation of these models. Because uncertainties are associated with regularization parameters, extra conditions are usually required to determine proper parameters for assessing inverse models. Commonly used techniques for assessment of a geophysical inverse model derived, generally iteratively, from a linear system are based on calculating the model resolution and the model covariance matrices. Because the model resolution and the model covariance matrices of the regularized solutions are controlled by the regularization parameter, a direct assessment of inverse models using only the covariance matrix may provide incorrect results. To find a proper regularization parameter, we consider an objective function that is the trace of a weighted sum of model resolution and model covariance matrices in the vicinity of a regularized solution where the linearity of inverse problems is normally held. Using the singular value decomposition, we derive explicit formulas to calculate a regularization vector and a weighting vector that result in a minimized objective function. With the optimum regularization vector and weighting vector, we obtain a trade-off solution between model resolution and model covariance in the vicinity of a regularized solution. The unit covariance matrix can then be used to calculate an error bar of the inverse model. We apply these formulas to inverse models of both synthetic and real surface-wave data.