# DIVA method

## Formulation

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

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

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).

### Parameter meaning

Using a characteristic length scale L, we introduce non-dimensional space coordinates (denoted by &\tilde) as follows:

• for the gradients: $\tilde{\boldsymbol{\nabla}} = L \boldsymbol{\nabla}$,
• for the domain: $\Omega = L^2 \tilde{\Omega}$.

The previous formula becomes:

$\tilde{J}[\varphi]=\sum_{i=1}^{N}\mu_i L^2 [d_{i}-\varphi(x_{i},y_{i})]^{2}+ \int_{\tilde{\Omega}}\left( \tilde{\boldsymbol{\nabla}}\tilde{\boldsymbol{\nabla}}\varphi : \tilde{\boldsymbol{\nabla}}\tilde{\boldsymbol{\nabla}}\varphi + { \alpha_{1} L^2} \tilde{\boldsymbol{\nabla}}\varphi \cdot \tilde{\boldsymbol{\nabla}}\varphi + \alpha_{0} L^4 \varphi^{2} \right) d \tilde{\Omega}.$

• 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 $\epsilon_i^2$ of data point i and the integral norm representative of the background field variance σ2, then the data weights are given by

$\mu_i L^2= 4 \pi \frac{\sigma^2}{\epsilon_i^2}.$

• 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 $\lambda = \frac{\sigma^2}{\epsilon^2},$ with a global noise level on data $\epsilon^2=\sum_i \epsilon_i^2/N,$, we can define relative weights on the data with noise $\epsilon_i^2$ as

${w}_i= {\mu_i L^2 \over {4 \pi \lambda} } \quad \textrm{with}\quad \sum_i{\frac{1}{w_i}} = N.$

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.

### 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 $\varphi_{e}$ is expanded in terms of connector values $\mathbf{q}$, which ensures the solution is continuously derivable, and shape functions $\mathbf{s}$, which serve to compute the field at any desired location:

$\varphi_{e} = \mathbf{q}^{T}_{e} s.$

Triangular finite-element mesh generated around the Sulu Sea.