Modeling uncertainty for future prediction requires drawing multiple posterior models. Such drawing within a Bayesian framework is dependent on the likelihood (data-model relationship) as well as prior distribution of the model variables, For the uncertainty assessment in the Earth models, we propose the framework of Modeling Uncertainty in Metric Space (MUMS) to achieve this in a general way. MUMS constructs a metric space where the models are represented exclusively by a distance correlated with or equal to the difference in their responses (application-tailored distance). In the framework of MUMS, various operations are available: projection of metric space by multi-dimensional scaling, model expansion by kernel Karhunen-Loeve expansion, generation of additional prior model by solving the pre-image problem, and generation of multiple posterior models by solving the post-image problem. We propose a robust solution for the pre-image problem: geologically constrained optimization, which utilizes the probability perturbation method from the solution of the fixed-point iteration algorithm. Additionally, we introduce a so-called post-image problem for obtaining the feature expansion of the ''true Earth'' by defining a distance as the difference in their responses. The combination of geologically constrained optimization and the post-image problem efficiently generates multiple posterior Earth models constrained to prior geologic information, hard data, and nonlinear time-dependent data. The proposed method provides a realistic uncertainty model for future prediction, compared with the result of the rejection sampler. We also propose a metric ensemble Kalman filter (Metric EnKF), which applies the ensemble Kalman filter (EnKF) to the parameterizations by the kernel KL expansion in metric space. Metric EnKF overcomes some critical limitations of EnKF: it preserves prior geologic information; it creates a stable and consistent filtering. However, the results of Metric EnKF applied to various cases including the Brugge field-scale synthetic reservoir show the same problem as with the EnKF in general, that is, it does not provide a realistic uncertainty model.