Let $X$ be a smooth manifold without boundary, and let $u \in \mathcal{D}'(X)$. Let $\Psi^0_{\mathrm{prop}}(X)$ denote the class of properly supported pseudodifferential operators of order $0$ on $X$, initially defined as maps $A:C_c^\infty(X) \to C^\infty(X)$ and extended by proper support to continuous linear maps $A:\mathcal{D}'(X)\to \mathcal{D}'(X)$. For $A \in \Psi^0_{\mathrm{prop}}(X)$, let $\operatorname{Ell}(A) \subset T^*X \setminus 0$ denote the conic elliptic set of $A$, and define

\begin{align*} \operatorname{Char}(A) := (T^*X \setminus 0) \setminus \operatorname{Ell}(A). \end{align*}

Assume the standard microlocal facts that ellipticity of a properly supported pseudodifferential operator at $p \in T^*X \setminus 0$ and smoothness of $Au$ imply $p \notin \operatorname{WF}(u)$, and that for every $p \notin \operatorname{WF}(u)$ there exists $A \in \Psi^0_{\mathrm{prop}}(X)$ elliptic at $p$ with operator wave front set contained in a conic open neighbourhood disjoint from $\operatorname{WF}(u)$. Then

\begin{align*} \operatorname{WF}(u)=\bigcap\left\{\operatorname{Char}(A): A \in \Psi^0_{\mathrm{prop}}(X) \text{ and } Au \in C^\infty(X)\right\}. \end{align*}