The Stochastic Calculus Reformulation of Data Assimilation: on Scale
Other literature type
(issn: 1607-7946, eissn: 1607-7946)
The understanding of uncertainties in Earth observations and simulations has been hindered by the spatial scale problem. In addition, errors caused by spatial scale change are an important part of uncertainty in data assimilation (DA). However, these uncertainties exceed the abilities of current theory. We attempted to address these problems. First, measure theory was used to propose a mathematical definition such that spatial scale is the function output of a measure given that its referential element and representative region are confirmed, and then the Jacobian matrix was used to describe the change of scale. Second, the scale-dependent variable was defined to further consider the heterogeneities. Last, under the Bayesian framework of DA, the scale-dependent uncertainty was studied based on stochastic calculus. The result formulated the scale-dependent error in DA. If we restrict the scale to a one-dimensional variable, the variation range of this type of error is proportional to the scale gap. Furthermore, assuming the observation operator is stochastic, we developed an example by introducing the stochastic radiative transfer equation. The new methodology will extend the recognition of the uncertainty in DA and may be able to address the scale problem.