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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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© Author(s) 2016. This work is distributed
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Research article
30 Jun 2016
Review status
A revision of this discussion paper is under review for the journal Nonlinear Processes in Geophysics (NPG).
The Stochastic Calculus Reformulation of Data Assimilation: on Scale
Feng Liu1,3 and Xin Li1,2 1Key Laboratory of Remote Sensing of Gansu Province, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, 730000, China
2Center for Excellence in Tibetan Plateau Earth Sciences, Chinese Academy of Sciences, Beijing 100101, P. R. China
3University of Chinese Academy of Sciences, Beijing 100049, P. R. China
Abstract. 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.

Citation: Liu, F. and Li, X.: The Stochastic Calculus Reformulation of Data Assimilation: on Scale, Nonlin. Processes Geophys. Discuss., doi:10.5194/npg-2016-35, in review, 2016.
Feng Liu and Xin Li
Feng Liu and Xin Li
Feng Liu and Xin Li


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Short summary
This is the first mathematical definitions of the spatial scale and its change. A definition of variable with respect to scale was also provided by considering the variation of geographical parameters. The stochastic calculus for data assimilation discovered the formulation of error caused by spatial scale. The results improve the ability to understand the uncertainty and scale problem in Earth observation, modeling and data assimilation.
This is the first mathematical definitions of the spatial scale and its change. A definition of...