Background Error Correlation Modeling with Diffusion Operators - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 172

Chapter 8
Background Error Correlation Modeling
with Diffusion Operators
Max Yaremchuk, Matthew Carrier, Scott Smith, and Gregg Jacobs
Abstract Many background error correlation (BEC) models in data assimilation
are formulated in terms of a positive-definite smoothing operator B that is employed
to simulate the action of correlation matrix on a vector in state space. In this
chapter, a general procedure for constructing a BEC model as a rational function
of the diffusion operator D is presented and analytic expressions for the respective
correlation functions in the homogeneous case are obtained. It is shown that this
class of BEC models can describe multi-scale stochastic fields whose characteristic
scales can be expressed in terms of the polynomial coefficients of the model.
In particular, the connection between the inverse binomial model and the well-
known Gaussian model B g D exp D is established and the relationships between
the respective decorrelation scales are derived.
By its definition, the BEC operator has to have a unit diagonal and requires
appropriate renormalization by rescaling. The exact computation of the rescaling
factors (diagonal elements of B ) is a computationally expensive procedure, therefore
an efficient numerical approximation is needed. Under the assumption of local
homogeneity of D , a heuristic method for computing the diagonal elements of
B is proposed. It is shown that the method is sufficiently accurate for realistic
applications, and requires 10 2 times less computational resources than other meth-
ods of diagonal estimation that do not take into account prior information on the
structure of