Outline (2009/5/9 e-mail from Aimé)
The 3-month goal is to write the software to enable a horizontal 2D wavelet-based method as an alternative to your current recursive-filter method, including appropriate diagnostics for comparing the methods.
The wavelets would also provide an alternative to horizontal global-spectral transforms in a limited area or on the sphere (we should fix either domain for the initial work).
In the second 3-month period we would conduct the comparison between old and new methods, evaluate which new topic(s) appear(s) potentially most to benefit from wavelets, and conduct a study to quantify that benefit. We mentioned roughly 5 candidate topics more or less concretely:
1. replacing a binary (was it land-nonland or cloud-noncloud?) mask with a wavelet-based mask in the ACAPS (I could not find a definition of "ACAPS" except "AFWA Coupled Simulation and Prediction System" e.g., p. 23 of www.jcsda.noaa.gov/documents/meetings/AdvPanel200901/AFOverview200901.pdf ---is that what you meant?)
2. using wavelets to decompose the transform to balanced fields to allow for location-and-scale dependent formulations of balance i.e., simple balance formulations at large scales or smooth-flow regions could be augmented by additional balance terms at small scales or more non-linear flow regions.
3. use wavelets to efficiently incorporate anisotropic contributions to the BE.
4. use wavelet thresholding to obtain a smooth field from noisy observations.
5. use wavelet adaptivity to enhance assimilation of fields with strongly localized multiscale features such as fronts.
Technical details (5/11 reply from Yann)
to have a horizontal 2D wavelet-based transform enabled to compare with the current (or improved, as I'm working on it) recursive-filter method. To make it clear, the proposed work has two different - but linked technical parts:
1. "gen_be" part
Data assimilation schemes represent the B matrix as a sequence of transform operators representing the square root operator B^1/2. Usually, people use some approximate errors samples (e.g. NMC method) to calibrate those operators. For instance, the output of "gen_be" contains some regression coefficients (balance transform), eigenvectors and eigenvalues of vertical background covariance matrix (vertical transform) and lengthscales (horizontal transform). This part does not include explicitly the associated transforms, which are rather embedded in the data assimilation scheme.
2. WRFVAR data assimilation part
The data assimilation schemes reads the output of "gen_be" that define the successive transforms. It applies it to a state vector within the minimization algorithm that produces the analysis from the background and from the observations. The successive linear transforms have to be coded twice, as the minimization requires the adjoint of those transforms. The adjoint transforms are applied to the gradient of the cost function.
The goal of the six-months work is to introduce a horizontal 2D wavelet-based transform . This is the last transform in the sequence that define B^1/2 (e.g. we would keep the balance and the vertical transforms as they are), and it would have to be introduced in both "gen_be" and WRFVAR parts to allow comparison with the current recursive filters transform. The comparison would include for example single observation experiments, and then data assimilation experiments with real observations.
I would suggest the following working plan/objectives
Months 1-3
Scientific depiction of the horizontal 2D wavelet-based transform
Technical implementation of the horizontal 2D wavelet-based transform in "gen_be" and in WRFVAR
Months 4-6:
Test of single observations over various domains
Depiction of the possibilities and limitations of horizontal 2D wavelet transforms to represent the grid-point anisotropies (c.f. Deckmin and Berre MWR 2005) and the power spectrum structure of the correlations (c.f. Bannister QJRMS 2008)
Data assimilation experiments over various domains
Scientific reflexion on possible extensions (vertical part, balance, curvelets...) related to the 5 topic points we mentioned.
I would consider the deliverable of a horizontal 2D wavelet-based transform to be the key point within the 6-month work, as it could be easily illustrated and shown in some ACAPS-related proposal.