# DIVA method

### From GHER

## Contents |

## Formulation

We are looking for the field which minimizes the cost function:

with

where

- α
_{0}penalizes the field itself (anomalies), - α
_{1}penalizes gradients (no trends), - α
_{2}penalizes variability (regularization), - ยต penalizes data-analysis misfits (objective).

Without loss of generality we can chose α_{2}=1 (homogeneous function).

### Parameter meaning

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:

- α
_{0}fixes the length scale*L*the first and the last term of the integral have a similar importance:

α_{0}*L*^{4} = 1.

- The terms μ
_{i}*L*^{2}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 σ^{2}, then the data weights are given by

- Finally α
_{1}fixes the influence of gradients:

α_{1}*L*^{2} = 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 μ_{i}, α_{0} and α_{1}, we can therefore work with relative weights *w*_{i} on data, an overall signal-to-noise ratio λ, a length scale *L* and a shape parameter ξ.

### Parameter determination

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.

## Solver

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: