We analyze, through linear algebra, the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics. To a first-order approximation, the local data misfit function in any inverse problem is valley-shaped and elongated in the directions of the null space of the Jacobian and/or in the directions of the smallest singular values. In nonlinear inverse problems, valleys persist; however, local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the Gauss-Newton approximation to the Hessian, the regularization term designed to provide convexity to the misfit function, or to noise in the data. Furthermore, noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder (in the case of existence). Understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise, incomplete sampling, regularization, and more fundamentally, simplified physical models.

The mining industry relies heavily on empirical analysis for design and prediction. An empirical design method, called the critical span graph, was developed specifically for rock stability analysis in entry-type excavations, based on an extensive case-history database of cut and fill mining in Canada. This empirical span design chart plots the critical span against rock mass rating for the observed case histories and has been accepted by many mining operations for the initial span design of cut and fill stopes. Different types of analysis have been used to classify the observed cases into stable, potentially unstable and unstable groups. The main purpose of this paper is to present a new method for defining rock stability areas of the critical span graph, which applies machine learning classifiers (support vector machine and extreme learning machine). The results show a reasonable correlation with previous guidelines. These machine learning methods are good tools for developing empirical methods, since they make no assumptions about the regression function. With this software, it is easy to add new field observations to a previous database, improving prediction output with the addition of data that consider the local conditions for each mine.