Introduction

Thus far we have considered models in which the future behavior of the system can be determined using equations and boundary conditions known at some initial time. In many Earth surface systems, however, the future behavior of the system cannot be predicted with certainty, either because the system behavior is sensitive to small-scale processes that cannot be fully resolved and/or because there is significant uncertainty in model input parameters. The climate system is a good example of a system that depends on small-scale processes (i.e. turbulence) that cannot be fully resolved in any numerical model. As a result, climate and weather models are inherently limited in their ability to predict the details of future climate and weather patterns. Soil permeability is a good example of a model input parameter with significant uncertainty. Soil permeability depends on the detailed structure and composition of the soil at a range of spatial scales, making an exact determination of permeability very difficult over large areas. As such, numerical models that require soil permeability as an input (e.g. models for runoff, infiltration, aquifer recharge, etc.) are limited in their precision no matter how finely they resolve the underlying physics of the problem.

In some applications where deterministic models cannot predict the future behavior of a system precisely, stochastic models can be useful for understanding the range of possible system behaviors.