In Getzler's famous paper "Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem", he states that for a (trace-class) pseudo-differential operator $P$ on a Riemannian manifold $M$, one has the formula $$ \mathrm{Tr} P = \int_{T^*M} \sigma(P), $$ where $\sigma(P)$ is the full symbol of $P$, which is calculated using "the exponential map of $M$ to pull back the pseudodifferential operator near $x \in M$ to $T_xM$ and calculate its symbol on $T^*_xM$ using the Euclidean structure" (quote of p.164).

However, as far as I know, the full symbol of a $\Psi$DO can be defined using the Levi-Civita connection, but only up to a smoothing symbol. And of course in applications, $P$ is usually a smoothing operator.Also, I could not get my hands on a full copy of the paper of Widom ("A complete symbolic calculus for pseudodifferential operators") cited there, and the preview of it on google books does not seem to help me.

So, my question is: How does one define $\sigma(P)$ to make the above formula true?