# Diva intro

In oceanography a typical concern consists in determining a field on a regular grid of positions r knowing N;d data in locations rj, j=1,..., N;d. This is called the gridding problem and is useful for many applications such as data analysis, graphical display, forcing or initialization of a model.

DIVA stands for Data-Interpolating Variational Analysis and is the implementation of Variational Inverse Method. It is designed to solve 2-D differential or variational problems of elliptic type with a finite element method. Its end is to obtain a gridded field from the knowledge of sparse data points.

### Formulation

We are looking for the field which minimizes the variational principle:

$J \left[\varphi\right] =\sum_{j=1}^{Nd}\mu_{j}\left[d_{j}-\varphi(x_{j},y_{j})\right]^{2}+ \| \varphi\| ^{2}$

with

$\|\varphi\|=\int_{D}( \alpha_{2} \boldsymbol{\nabla}\boldsymbol{\nabla}\varphi : \boldsymbol{\nabla}\boldsymbol{\nabla}\varphi +\alpha_{1} \boldsymbol{\nabla}\varphi \cdot \boldsymbol{\nabla}\varphi + \alpha_{0} \varphi^{2})\, d D$

where

• α0 penalizes the field itself (anomalies),
• α1 penalizes gradients (no trends),
• α2 penalizes variability (regularization),
• ยต penalizes data-analysis misfits (objective).

Without loss of generality we can chose α2=1 (homogeneous function).