This paper presents a new high-order, non-stationary sequential simulation approach, aiming to deal with the typically complex, curvilinear structures and high-order spatial connectivity of the attributes of natural phenomena. Similarly to multipoint methods, the proposed approach employs spatial templates and a group of training images (TI). A coarse template with a fixed number of data points and a missing value in the middle is used, where the missing value is simulated conditional to a data event found in the neighborhood of the middle point of the template, under a Markovian assumption. Sliding the template over the TI, a pattern database is extracted. The parameters of the conditional distributions needed for the sequential simulation are inferred from the pattern database considering a set of weights of contribution given for the patterns in the database. Weights are calculated based on the similarity of the high-order statistics of the data event of the hard data compared to those of the training image. The high-order similarity measure introduced herein is effectively invariant under all linear spatial transformations. Following the sequential simulation paradigm, the template chosen is sequentially moved on a raster path until all missing points/nodes are simulated. The high-order similarity measure allows the approach to be fast as well as robust to all possible linear transformations of a training image. The approach respects the hard data and its spatial statistics, because it only considers TI replicate data events with similar high-order statistics. Results are promising.