  Let $\mathcal{G}$ be a locally compact \'{e}tale groupoid and
$\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators
on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a
notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$,
generalising the metric space case introduced by Roe. Our main result shows
that when $\mathcal{G}$ is additionally $\sigma$-compact and amenable, an
equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced
groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local.
This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$
using coarse geometry. Our main tool is a description for operators in
$\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to
the computer tomography. As applications, we recover a result by \v{S}pakula
and the second-named author in the metric space case, and deduce new
characterisations for reduced crossed products and uniform Roe algebras for
groupoids.
