To model random fields whose variability changes at differing scales we use multiscale kernel convolution models that rely on nested grids of knots at different resolutions. Thus, lower order terms capture large scale features, while high order terms capture small scale ones. To accommodate the space-varying nature of the variability we consider models where the coefficients of the kernel expansion vanish adaptively as the resolution grows. We develop an approach that relies on knot selection, to achieve parsimony, and discuss how this induces a field with spatially varying resolution. We extend shotgun stochastic search to the multi resolution model setting. Alternatively, we consider maximization methods for the estimation of the optimal model. These consist of using a sparsity inducing regularization term that, using an overlapping group penalty incorporates information about the structure of a recursive partitioning of the domain. We demonstrate that the proposed methods are computationally competitive and produce excellent fit to both synthetic and real datasets.