DIVA method

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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),
  • µ 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.
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