Let $\pi:E\to M$ be a rank $k$ vector bundle over a compact manifold $M$. The usual method to associate a sphere bundle to $E$ is by considering only vectors of length 1 in each fiber of $E$ (after choosing a metric on the bundle). This yields a bundle $S(E)\to M$ with fiber $S^{k-1}$.
My question is: Can we construct a $k$-sphere bundle $C(E)\to M$ from $E$ by looking at the one-point compactification of each fiber of $E$?
If this is indeed possible some details to the construction and references would be appreciated.
I suppose that the zero section of $E\to M$ would induce a section of $C(E)\to M$. This construction is probably related to the construction of the Thom-space, where the one-point compactification of the total space $E$ is considered.