Implementing multi-scale background error covariance matrix

[hga007]

Description

This PR implements a multi-scale background error covariance (B) operator into ROMS 4D-Var. It is based on the formulation of Weaver et al. (2013, 2016, 2018). It is activated with the MULTI_SCALE_B option. The new formulation uses a normalized implicit diffusion operator that represents Matérn-class correlation functions, which are solved using the Chebyshev iterations algorithm (Weaver et al., 2016, 2018).

For technical information on its implementation in ROMS, please see the multi_scale_B.pdf document.

A secondary independent effect of this development is to enable spatially varying correlation scales in modeling the background-error covariance via convolutions of pseudo-diffusion operators, which also use the Chebyshev iteration solver. It allows correlation functions (Whittle–Matérn family) with more complex shapes. The implicit diffusion operator is more efficient for multiple correlation length scales. It is up to the user to determine such spatially varying horizontal correlation length scales. For example, they can be computed from the horizontal distribution of the first Rossby radius of deformation.

References

Weaver, A.T. and I. Mirouze, 2013: On the diffusion equation and its application to isotropic and anisotropic correlation modeling in variational assimilation, Q. J. R. Meteorol. Soc., 139, 242-260, doi:10.1002/qj.1955.

Weaver, A.T., Tshimanga, J., and A. Piacentini, 2016: Correlation operators based on an implicitly formulated diffusion equation solved with the Chebyshev iteration, Q. J. R. Meteorol. Soc., 142, 455-471, doi: 10.1002/qj.2664.

Weaver, A.T., Gürol, S., Tshimanga, J., Chrust, M., and A. Piacentini, 2018: "Time"-parallel diffusion-based correlation operators, Q. J. R. Meteorol. Soc., 144, 2067-2088, doi: 10.1002/qj.3302.

Issue(s) addressed

None.

Impacts

Please describe any impacts this change may have on performance or other parts of the code.

Dependencies

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Checklist

• I have reviewed code updates or enhancements described in this PR
• I have run and tested the changes before creating the PR