Direct current resistivity method: Data acquisition, forward modeling and inversion

To reduce the null space of a model, more data are collected according to the analysis based on previous data. This process is repeated until the null space of a model can not be further reduced given a noise level of the data. The system is linear, or can be linearized. The null space is found by the singular value decomposition (SVD) analysis of the sensitivity matrix, Suppose that there are a large number of redundant data we can potentially collect, this adaptive data acquisition approach collects much less redundant data to get an inversion result which can be got from all possible data. This approach is efficient especially when there are potentially unlimited number of data. An example is given based on direct current resistivity (DC) resistivity using one-dimensional (1D) model.; Given a total of P electrodes, the dimension of the data space for dipole-dipole DC resistivity measurements is shown as [P*(P-3)]/2 + 1. One of bases of the data space is given. Data independence analysis among commonly used dipole-dipole measurement arrays, known as "skips", is given.; The standard approach in dipole-dipole DC resistivity forward modeling is to find the potentials resulting from a current dipole by solving a set of linear equations for this dipole. For N electrodes, the number of sets of linear equations to solve can be as large as N*(N-1)/2. For large numbers of current dipoles, this approach is not computationally efficient. By first computing only single current source (pole) data, then combining these data into dipole-dipole data, the maximum number of sets of linear equations that need to solved is reduced to N. Compared to conventional methods, this method is much more efficient in cases where we have large numbers of different dipoles. Comparison between this approach and the standard approach shows no loss of accuracy with computing speeds increased by up to 90%.; To address the non-uniqueness of a model, regularization terms are normally used during the inversion process to find the best model based on a priori information or model assumptions. One kind of most commonly used regularization is the smoothness constraint. However, if sharp resistivity contrasts exist within the earth, inversions with smoothness constraints may fail to converge or may yield suboptimal results. As the presence or location of sharp contrast boundaries is normally unknown, it is in general impossible to apply the optimal constraints prior to the inversion. An algorithm is presented to locate the possible sharp contrast boundaries as part of the inversion through iterative updates of smoothness constraints and allow significant improvement of the final image of subsurface properties.