Diva intro

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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).

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