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Invariant correction terms

The shallow-water equations do not depend neither on the orientation nor on the origin of the frame in which the coordinates are expressed: they are invariant under the action of $ SE(2)$ , the Special Euclidean group of isometries of the plane $ \mathbb{R}^2$ . Consequently, functions $ F_h$ and $ F_v$ must be invariant under the action of $ SE(2)$ . Symmetries have been very recently introduced for observer design in [1,40] for engineering problems. The aim of this work is to consider correction terms that respect the underlying physics of the system.

To find the scalar term $ F_h$ , we use the standard result (see e.g. [98]), which states that any scalar differential operator invariant by rotation and translation writes $ Q(\Delta)$ , where $ Q$ is a polynomial and $ \Delta$ is the Laplacian. By considering the invariance by rotation for the vectorial velocity [87], we get the following family of scalar terms:

$\displaystyle F_h = Q_1(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-h)+\nabla\left( Q_2(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-h)\right).\hat{v}+f_h,$ (3.52)

where $ Q_1$ and $ Q_2$ are scalar polynomials in $ \Delta$ , and $ f_h$ is an integral term defined below. More precisely,

$\displaystyle Q_i(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-h)=\sum_{k=0}^N a_k^i(h,... ...}\vert^2,\hat{h}-h)\Delta^k\left(b_k^i(h,\vert\hat{v}\vert^2,\hat{h}-h)\right),$ (3.53)

where $ a_k^i$ and $ b_k^i$ are smooth scalar functions such that

$\displaystyle a_k^i(h,\vert\hat v\vert^2,0) = b_k^i(h,\vert\hat v\vert^2,0) = 0.$ (3.54)

For the vectorial correction term $ F_v$ , we use the vectorial counterpart:

$\displaystyle F_v=P_1(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-h)\hat{v}+\nabla \left( P_2(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-h)\right)+f_v,$ (3.55)

where $ P_1$ and $ P_2$ are polynomials in $ \Delta$ , like $ Q_1$ and $ Q_2$ .

Let us now find the integral terms $ f_v$ and $ f_h$ that are invariant by rotation and translation. They can be expressed as a convolution between the previous invariant differential terms and a two-dimensional kernel $ \psi(\xi,\zeta)$ . The previous terms being invariant by rotation, the value of the kernel should not depend on a particular direction, and so $ \psi$ must be a function of the invariant $ \xi^2+\zeta^2$ . The integral correction terms write:

$\displaystyle f_v(x,y,t)=\iint \left[ R_1(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-... ...t{h}-h)\right) \right]_{(x-\xi,y-\zeta,t)} \phi_v(\xi^2+\zeta^2)\, d\xi d\zeta,$ (3.56)

$\displaystyle f_h(x,y,t)=\iint \left[ S_1(\Delta,h,\vert\hat{v}\vert^2,\hat{h}-... ...right).\hat{v} \right]_{(x-\xi,y-\zeta,t)} \phi_h(\xi^2+\zeta^2)\, d\xi d\zeta,$ (3.57)

where $ R_i$ and $ S_i$ are defined like $ Q_i$ and $ P_i$ .

The support of $ \phi_v$ (resp. $ \phi_h$ ) is a subset of $ \mathbb{R}$ . Its characteristic size defines a zone in which it is significant to correct the estimation with the measurements. The integral formulation is actually quite general: if $ \phi_v$ and $ \phi_h$ are set equal to Dirac functions, one obtains the differential terms.

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