There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is the equivalent of the residue. Let me explain what I mean.

Let $(E, \nabla)$ be a vector bundle with connection on some smooth variety $X$ over a field $k$ of characteristic zero. Assume that $(E, \nabla)$ has regular singularities, meaning that there is a logarithmic extension $\bar{E} \to \bar{E} \otimes \Omega_{\bar{X}}(\log D)$, where $\bar{X}$ is a smooth compactification of $X$ such that $D=\bar{X}-X$ has normal crossings. Then, for each irreducible component of $D_i$, the composition with the Poincare residue gives a map $$ \bar{E} \to \bar{E} \otimes \mathcal{O}_{D_i} $$ which in fact restricts to an endomorphism of $\bar{E} \otimes \mathcal{O}_{D_i}$.

What is the $\ell$-adic analogue of this? Assume we are over a finite field and in the same situation as before ($X \hookrightarrow \bar{X}$ good compactification, this is not anymore automatic!). A regular singular connection should be replace by a representation $\pi_1^{et}(X) \to GL_n(\bar{\mathbb{Q}}_\ell)$ tamely ramified at $D$. How do I get the residue?

One of my concerns is that, in order to define the residue, I need to choose a logarithmic extension and the residue *does depend* on this choice. What is the substitute of the logarithmic extension in the $\ell$-adic context?