Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.

Let $\rho:G \to GL_d(\mathbf Q_\ell)$ be a Galois representation.

Is there a useful notion of "unramified" at $x$, where $x$ is a point of $\mathbf P^1_{\mathbf C}$?

I'm trying to mimick the arithmetic situation (replacing $\mathbf C(t)$ by $\mathbf{Q}$ and $x$ by a prime number $p$), but this gives me that $\rho$ is unramified everywhere. Does anybody have any suggestions? I'd really be interested in knowing whether there is a useful notion of ramification in this setting.

Here's what I would like such a notion of ramification to satisfy.

Let $X$ be a smooth projective variety (say of general type) over $\mathbf{C}(t)$ which has a smooth projective model over $\mathbf{A}^1$, but not over $\mathbf{P}^1$. Then, I would like the associated Galois representation (via etale cohomology of $X_{\overline{\mathbf C(T)}}$) to be unramified at all $x$ in $\mathbf{A}^1$ and to be ramified at $\infty$.