Let \begin{equation}
\mathcal{R}^h = \alpha_1 \mathcal{R}^h_1 + \alpha_2 \mathcal{R}^h_2
\end{equation} where \begin{align}
\mathcal{R}^h_1 &= \frac{1}{\text{area}(\Omega)} \int_\Omega ||A_\beta b||_2^2 \; \mathrm{d}m, \\ \mathcal{R}^h_2 &= \frac{1}{\text{area}(\Omega)} \int_\Omega ||A_\beta A_\beta^\intercal b||_F^2 \; \mathrm{d}m.
\end{align}

Above, $||\cdot||_2$ denotes the Euclidean norm and $||\cdot||_F$ denotes the Frobenius norm (https://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm). $A_\beta$ is the "anisotropy operator" \begin{equation}
A_\beta = \begin{pmatrix} \nabla_{\hat{\vec{u}}} \\ \sqrt{\beta} \nabla_{\hat{\vec{u}}^\perp} \end{pmatrix}
\end{equation} where $\nabla_{\hat{\vec{u}}}$ denotes the directional derivative in the direction of flow $\hat{\vec{u}}$, the hat $\hat{\;}$ denotes a unit vector, $\hat{\vec{u}}^\perp$ denotes the unit vector perpendicular to $\hat{\vec{u}}$, and $\beta \in (0, 1]$ is a number which determines the amount of anisotropy ($\beta=1$ means isotropic).

The integrand of $\mathcal{R}^h_1$ is then (up to a constant factor) the same as in Equation 5 from doi.org/10.5194/egusphere-2026-788 (and, if the convexity weight $\gamma=0$, Equation 10 from doi.org/10.1017/jog.2022.41): \begin{align}
||A_\beta b||_2^2 &= |\nabla_{\vec{u}}b|^2 + \beta |\nabla_{\vec{u}^\perp}b|^2 \nonumber \\ &= |\vec{u} \cdot \nabla b|^2 + \beta |\vec{u}^\perp \cdot \nabla b|^2.
\end{align}

Writing the vector components as $\hat{\vec{u}} = (\hat{u}_x, \hat{u}_y)^\intercal$ and $\nabla b = (\partial b / \partial x, \partial b / \partial y)^\intercal = (b_x, b_y)^\intercal$, the integrand of $\mathcal{R}^h_2$ is \begin{align}
||A_\beta A_\beta^\intercal b||_F^2 &= \left\Vert \begin{pmatrix}
\nabla_{\hat{\vec{u}}}^2 b & \sqrt{\beta} \;\nabla_{\hat{\vec{u}}} \nabla_{\hat{\vec{u}}^\perp} b \\ \sqrt{\beta} \;\nabla_{\hat{\vec{u}}} \nabla_{\hat{\vec{u}}^\perp} b & \beta \; \nabla_{\hat{\vec{u}}^\perp}^2 b
\end{pmatrix} \right\Vert_F^2 \nonumber \\
&= |\nabla_{\hat{\vec{u}}}^2 b|^2 + 2\beta \; |\nabla_{\hat{\vec{u}}} \nabla_{\hat{\vec{u}}^\perp} b|^2 + \beta^2 \; |\nabla_{\hat{\vec{u}}^\perp}^2 b|^2 \nonumber \\
&= |\hat{\vec{u}} \cdot \nabla (\hat{\vec{u}} \cdot \nabla b)|^2 + 2 \beta |\hat{\vec{u}} \cdot \nabla (\hat{\vec{u}}^\perp \cdot \nabla b)|^2 + \beta^2 |\hat{\vec{u}}^\perp \cdot \nabla (\hat{\vec{u}}^\perp \cdot \nabla b)|^2 \\
&= |\hat{\vec{u}} \cdot \nabla (\hat{u}_x b_x + \hat{u}_y b_y)|^2 + 2 \beta |\hat{\vec{u}} \cdot \nabla (\hat{u}_y b_x - \hat{u}_x b_y)|^2 + \beta^2 |\hat{\vec{u}}^\perp \cdot \nabla (\hat{u}_y b_x - \hat{u}_x b_y)|^2 \nonumber \\
&= |\hat{u}_x^2 b_{xx} + 2\hat{u}_x\hat{u}_y b_{xy} + \hat{u}_y^2 b_{yy}|^2 + 2\beta|\hat{u}_x\hat{u}_y b_{xx} + (\hat{u}_y^2 - \hat{u}_x^2)b_{xy} - \hat{u}_x\hat{u}_y b_{yy}|^2 \nonumber \\ & \quad + \beta^2 |\hat{u}_y^2 b_{xx} -2\hat{u}_x \hat{u}_y b_{xy} + \hat{u}_x^2 b_{yy}|^2
\end{align}

This is invariant under rotation of the $x$ and $y$ axes.

Letting $\beta=1$, we obtain an expression independent of $\hat{\vec{u}}$: \begin{align}
||A_1 A_1^\intercal b||_F^2 &= |\hat{u}_x^2 b_{xx} + 2\hat{u}_x\hat{u}_y b_{xy} + \hat{u}_y^2 b_{yy}|^2 + 2|\hat{u}_x\hat{u}_y b_{xx} + (\hat{u}_y^2 - \hat{u}_x^2)b_{xy} + \hat{u}_x\hat{u}_y b_{yy}|^2 \nonumber \\ & \quad + |\hat{u}_y^2 b_{xx} -2\hat{u}_x \hat{u}_y b_{xy} + \hat{u}_x^2 b_{yy}|^2 \nonumber \\
&= \cdots \nonumber \\
&= (\hat{u}_x^2 + \hat{u}_y^2)^2 \left[ b_{xx}^2 + 2b_{xy}^2 + b_{yy}^2 \right] \nonumber \\
&= b_{xx}^2 + 2b_{xy}^2 + b_{yy}^2.
\end{align}