We are looking for the field which minimizes the cost function:
- α0 penalizes the field itself (anomalies),
- α1 penalizes gradients (no trends),
- α2 penalizes variability (regularization),
- µ penalizes data-analysis misfits (objective).
Without loss of generality we can chose α2=1 (homogeneous function).
Using a characteristic length scale L, we introduce non-dimensional space coordinates (denoted by &\tilde) as follows:
- for the gradients: ,
- for the domain: .
The previous formula becomes:
- α0 fixes the length scale L the first and the last term of the integral have a similar importance:
α0L4 = 1.
- The terms μiL2 fix the weights on the individual observations. If the typical misfit is represented by the observational noise standard deviation of data point i and the integral norm representative of the background field variance σ2, then the data weights are given by
- Finally α1 fixes the influence of gradients:
α1L2 = 2ξ.
he default value in Diva is ξ = 1.
With the overall signal-to-noise ratio defined as with a global noise level on data , we can define relative weights on the data with noise as
Instead of the original parameters μi, α0 and α1, we can therefore work with relative weights wi on data, an overall signal-to-noise ratio λ, a length scale L and a shape parameter ξ.
The method aims to provide objective tools to estimate the analysis parameters L and λ:
- the correlation length L is determined by a fit of the data correlation onto a theoretical function.
- the signal-to-noise ratio λ is estimated using a generalized-cross validation.
The minimization of the cost function is performed with a finite-element method: the domain of interest (i.e., the considered sea). Over each element e, the solution is expanded in terms of connector values , which ensures the solution is continuously derivable, and shape functions , which serve to compute the field at any desired location: