Inversion of potential fields on nodes for large grids

• The new approach of stochastic inversion on nodes reduces the memory requirement.
• There is no limit on the number of discretizing prisms.
• Point density measurements are easily included in the inversion as known nodes.
• The nodes are related to the prisms in a geostatistically fully coherent manner.
• Physics is respected: gravity anomaly computed from inverted densities match gravity data

The non-iterative direct inversion of potential field data by stochastic approach enables to incorporate in a coherent way a priori geological knowledge, the known densities on any support size and the gravity data. The weakness of the method is the necessary computation of the parameter covariance matrix. For a large mesh made of prisms, the matrix is simply too large to fit in memory. The new approach approximates the prism covariance matrix by a surrogate matrix computed from the covariance matrix of a reduced set of nodes aimed at representing the whole domain of inversion. Care is taken to preserve the properties of direct stochastic inversion on the whole set of prisms. Hence, the approach accounts in a consistent way for the support effect, the inversion remains exact in the absence of noise on data, point and block known densities are exactly reproduced, any set of linear constraints can be applied, and the inversion is non-iterative in all cases. It is shown on synthetic examples that the number of nodes needs not to be very large to ensure a good approximation of the parameter covariance matrix or to ensure similarity of the inversion solutions. An application to a gravity survey including borehole density data shows the applicability of the method for a large number of cells in the inversion domain. Even with as much as 10,000 nodes and one million prisms, the computing time remained acceptable at less than two hours on a workstation. The inverted solution obtained with the nodes approach is compared to a direct kriging of borehole density data and to direct inversion using only the gravity data. The solution combining both information is different from the inversion using only gravity, but only in the area where borehole data are numerous. Although shown for the gravity–density potential, the approach is general and can be extended to magnetic-susceptibility and joint inversion. The proposed approach helps solving the recurrent problem of the application of stochastic inversion to large grids.